Performance analysis for T.H.E.M.I.S(*) image stabilizer ...230 G. Molodij and J. Rayrole:...

ASTRONOMY & ASTROPHYSICS FEBRUARY II 199, PAGE 229

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Astron. Astrophys. Suppl. Ser. 128, 229-244 (199)

Performance analysis for T.H.E.M.I.S(*) image stabilizeroptical system

II. Anisoplanatism limitations(*)Telescope Heliographique pour l’Etude du Magnetisme et desInstabilites de l’atmosphere Solaire

G. Molodij and J. Rayrole

Observatoire de Paris URA 326, CNRS, 61 Av. de l’Observatoire, 75014 Paris, France

Received March 13; accepted July 1, 1997

Abstract. Numerical simulations of anisoplanatism ef-fects after compensation by the T.H.E.M.I.S. image stabi-lizer optical system when observing extended sources arepresented. The residual wavefront error after correctionis computed using the expansion on the Zernike polyno-mials to analyze the performances of different adaptiveoptics systems for solar observations. The long exposureoptical tranfer functions are derived to simulate the im-age quality after correction for extended field of view. Wedemonstrate the capacity of the image stabilizer systemfor very large field of view observations with medium im-age quality while the capacity of high order adaptive opticssystem is suitable for high image quality observations butto the price of a reduction of the field of view.

Key words: telescopes — atmospheric effects —techniques: miscellaneous — Sun: granulation

1. Introduction

The franco-Italian T.H.E.M.I.S. telescope (T.H.E.M.I.S.is a French acronym which stands for “solar heliographictelescope for the study of solar magnetism and atmo-spheric instabilities”) is designed to obtain very precisemeasurements of the solar magnetic field vector from si-multaneous observations of different polarized spectrallines. Image stabilization and high image quality are re-quirements for the observation of small solar magnetic fea-tures like magnetic flux tubes, so that the light passingthrough the entrance of the spectrograph slit will comefrom the same solar structure troughout the observationtime (' 0.4 second) which is longer than the caracteris-tic time of evolution of the terrestrial atmospheric tubu-lence. One of the main optical elements integrated into

T.H.E.M.I.S. optical train is a tiltable mirror which cor-rects instrument vibrations and steering as well as ran-dom wavefront caused by atmospheric turbulence (Rayrole1992).

Solar physics research requires observations of mag-netic features in a range from one tenth of an arcsecond,for flux tubes, to a few arc-minutes, for active regions.Another demand on the image compensation system isthat the observer should be able to analyze a large field ofview. So, an estimate of image quality after correction us-ing the T.H.E.M.I.S. image stabilizer system is presented:1) To analyze the compensation degradation by adaptiveoptics system across large fields of view taking into ac-count the effects of anisoplanatism.2) To determine the usable limited field of view of thewavefront analysis corresponding to the isoplanatic patch.Anisoplanatism poses a severe problem if the field of viewof the correlation tracker is not cut down to approxi-mately the isoplanatic patch. The new tracking methodcalled granulation tracking described in a previous paper(Molodij et al. 1996) is devoted to measure image mo-tions with extended incoherent sources. The granulationtracker, incorporated in the T.H.E.M.I.S. optical arrange-ment is designed to work on a square field of view of thegranulation image which can be adjusted from 2 × 2 arc-seconds to 12 × 12 arcseconds, depending on the Friedparameter r0.

The favorable turbulence conditions at the Izana site inthe canary Island (r0 > 15 cm for 60% of the observationtimes) (Barletti et al. 1973) suggests that image stabiliza-tion system is well adapted for improving image resolutionof large field of view observations in visible wavelengthwith the use of a tiltable mirror. This is due to the factthat T.H.E.M.I.S. is a 0.9 meter class telescope (D/r0 <6 for 60% of the observation times). So, atmospheric

230 G. Molodij and J. Rayrole: Performance analysis for T.H.E.M.I.S image stabilizer optical system. II.

perturbations are esentially caused by the wavefront tiltswhose variances represent 90% of the total variance phase(Noll 1976).However, it must be keep in mind that any correlationtracker makes wavefront slope measurements in order tocorrect pure wavefronts tilts. Such problems are analyzedin this paper when observing extended sources.

A new approach to wavefront sensing on extended,complex objects based on the curvature sensing tech-nique while imaging the Sun has been recently proposedby R. Kupke, F. Roddier and D.L. Mickey (Kupke et al.1994). The goal of this paper is also to investigate andcompare the performances of image stabilizer optical sys-tem and adaptive optics system which will be able to cor-rect the first aberrated modes as tilt, focusing and astig-matism for the 0.9 meter telescope.

In order to evaluate the limitations due to angularanisoplanatism and slope measurements by correlationtrackers, this paper presents a theoretical analysis basedon the modal control of the adaptive optics system. Modalcontrol is an helpful tool in adaptive optics in order tomanage optimal correction in terms of temporal and an-gular decorrelation of turbulent wavefront (Rousset 1993;Gendron & Lena 1994). The analysis uses Mellin trans-form techniques to evaluate the effects of anisoplanatismupon the performance of adaptive optics system (Chassat1989; Sasiela 1994; Molodij & Rousset 1997).In Sect. 2, the residual wavefront distortions associatedare evaluated modally in terms of Zernike polynomialswhich are chosen for their simple analytical form and be-cause of the correspondance of the low-order Zernike poly-nomials to physically controlable modes of correction, .i.e,tilt, focusing, astigmatism, etc. The problem of the wave-front tilts correction from wavefront slope measurementsis considered in Sect. 3 using the modal expansion on theZernike polynomials (Primot et al. 1990).These analysis are directly related to the Zernike co-efficient angular correlations between two plane waves(Chassat 1992). Numerical results are presented in Sect. 4for the modelized turbulence profile C2

n resulting from ex-perimental measurements on the Izana site by Arcetri uni-versity (Barletti et al. 1973). In Sect. 5, the optical transferfunctions are evaluated for a Zernike expansion followingthe Wang and Markey approach (Wang & Markey 1978).The statistics of the Zernike coefficients for an expansionof Kolmogorov turbulence phase distortion have been de-rived by Noll (Noll 1976) and applied to calculate the op-tical transfer function (OTF ) for a point source (Wang& Markey 1978). We develop the calculation of OTF tocases when the observing source is extended in order toevaluate the degradation in the field of view of the im-age due to anisoplanatism after adaptive optics compen-sation. We compare the performances of image stabilizeroptical system and adaptive optics system which will cor-rect aberrations as tilt, focusing and astigmatism on solargranulation images.

2. Residual wavefront error

In order to determine the compensation quality in a field ofview, the residual phase variance over the telescope aper-ture is expressed expanding the turbulent wavefront on theZernike polynomials (or modes). The properties of Zernikepolynomials, denoted Zj , are well described by Noll (Noll1976) whose notation is adopted.The wavefront φ(r, α) incoming from a determined areaon the Sun, located at angular distance α from the on-axisobserved direction, is expanded as (piston mode removed):

φ(Rρ, α) =∞∑j=2

aj(α) Zj(ρ) (1)

where ρ is the normalized position vector in the aperture,R the radius of the telescope aperture and the aj(α) theZernike expansion coefficients given by:

aj(α) =

∫d2ρW (ρ) φ(Rρ, α) Zj(ρ) (2)

introducing the Noll normalisation function:

W (ρ) =

{1π : if |ρ| ≤ 10 : elsewhere.

(3)

So, the polynomials Zj(ρ) normalised on the telescopeaperture are defined in polar coordinates (ρ, θ) by:

Zj(ρ, θ) =√n+ 1

Rmn (ρ)

√2cos(mθ) : j even m 6= 0

Rmn (ρ)√

2sin(mθ) : j odd m 6= 0R0n(ρ) : m = 0

(4)

and

Rmn (ρ) =

n−m2∑s=0

(−1)s (n− s)!

s! [n+m2 − s]! [n−m2 − s]!

ρn−2s (5)

where n is the radial degree of the jth polynomial and mits azimuthal frequency.Let φ(Rρ) be the estimated wavefront reconstructed bythe adaptive optics system on the first J polynomials fromthe measurements. In this paper, noise in the wavefrontmeasurement will not be considered, therefore we write:

φ(Rρ) =J∑j=2

ajZj(ρ). (6)

The residual phase variance ε2(J, α) is estimated for anadaptive optics system having J degrees of freedom andfor an angular distance α between the two directions, cal-culating:

ε2(J, α) =

∫d2ρW (ρ) < (φ(Rρ, α)− φ(Rρ))2 > . (7)

G. Molodij and J. Rayrole: Performance analysis for T.H.E.M.I.S image stabilizer optical system. II. 231

Using the orthogonality properties of Zernike polynomials,Eq. (1) and Eq. (6), and the stationarity hypothesis, theresidual phase variance is:

ε2(J, α) =∞∑

j=J+1

< (aj)2 > + 2

J∑j=2

[< (aj)

2 >

− < aj(α)aj(0) >] . (8)

This result is classical in the estimation of wavefrontresidual error on the Zernike polynomial (Chassat 1989;Rousset 1993). The first term in brackets of the right handside of Eq. (8) represents the fitting error for an adaptiveoptics system which corrects the first J polynomials. Ithas been given by Noll (Noll 1976). The other term of theright hand side of the Eq. (8) is the spatial and angularwavefront errors.The residual error can be written as:

ε2(J, α) =∞∑j=2

< (aj)2 >

−J∑j=2

[2 < aj(α)aj(0) > − < (aj)

2 >]︸︷︷︸

A(j,α)

. (9)

The quantity A(j, α) is the spatial and angular error forthe corrected polynomial Zj . Correction by the adaptiveoptics system will be effective when A(j, α) is larger than0 for each polynomial Zj , i.e. when:

< aj(α)aj(0) >

< (aj)2 >≥ 0.5. (10)

When the criterion of Eq. (10) is satisfied, the measuredwavefront at angular distance α provides a satisfactory fitof the on-axis wavefront when the correlation criterion ofEq. (10) is verified for all the correction modes. For in-stance, this criterion allows to select the number of usefulcorrection modes for a given angular distance between thetwo wavefronts. Keeping in mind that all the low ordersmust be first corrected, a higher order mode will be cor-rected only if the criterion is verified. This result is generalin wavefront residual error estimation for adaptive opticsusing the modal analysis (Rousset 1993).

Now, the Zernike coefficient angular correlations< aj(α)aj(0) > between the two wavefront directions canbe derived as indicated in References (Valley & Wandzura1979; Chassat 1989; Molodij & Rousset 1997). Therefore,the Zernike coefficient angular correlation can be writtenas:

< aj(α)aj(0) > = 3.895 (n+ 1)

(D

r0

) 53

.

∫ Latm

0dhC2

n(h)In,m(αhR

)∫ Latm

0dhC2

n(h)(11)

where r0 in this equation is the Fried parameter, D isthe telescope diameter (D = 2R), C2

n is the turbulenceprofile, h is the altitude along the propagation path fromLatm (the effective altitude of the considered atmosphereis 20000 meter) to the telescope, and using the notationx = αh

R :

In,m(x) = sn,m

∫ ∞0

dK K−143 J2

n+1(K) [ J0(Kx)

+ kj J2m(Kx)] (12)

where K is the Fourier space of the Zernike polynomialsin polar coordinates (Noll 1976). and

kj =

{0 : if m = 0

(−1)j : if m 6= 0and

sn,m =

{1 : if m = 0

(−1)(n−m) : if m 6= 0.(13)

Let us notice that it is also possible to deduce fromEq. (11) and (12), the Zernike polynomial coefficient vari-ances. The result is in perfect agreement with the vari-ances given by Noll (Noll 1976). Let us remark that thevariance only depends on the radial degree n and is givenby:

< (aj)2 >= 3.895 (n+ 1)

(D

r0

) 53

In,0(0) (14)

where In,0(0) =∫∞O

dK K−14

3 J2n+1(K).

To calculate the residual phase variance Eq. (8), weconsider first that the number J of corrected polynomi-als is chosen in order to include all the polynomials ofthe maximum radial degree N . Knowing that there isn+ 1 polynomials per radial degree n, we have therefore:

J = (N+1)(N+2)2 .

Secondly, the azimuthal frequency symmetry properties ofZernike coefficient correlations are useful. Summation onall azimuthal frequencies of a radial degree leads to only aradial degree dependence. Then, we have using Eq. (11):

ε2(N,α)=3.895

(D

r0

) 53

[∞∑n=1

(n+ 1)2In,0(0)−N∑n=1

(n+ 1)2

{2

∫ Latm

0 dhC2n(h) In,m(αhR )∫ Latm

0 dhC2n(h)

− In,0(0)

}]· (15)

Integral In,m(x) (Eq. (12)) can be evaluated in closedform using Mellin transform techniques as proposed bySasiela and expressed in terms of rapidly converging series(Sasiela 1994). This integral is transformed into a Mellin-Barnes integral and is then expressed as a sum of gen-eralised hypergeometric functions in the appendix A. Asmentioned by Tyler (Tyler 1990), this analytic form is


related to integral involving product of three Bessel func-tions and can be evaluated only for restrictive cases, i.e.here depending on the angular separation. To overcomethis limitation, Chassat (Chassat 1992) proposed an an-alytic solution applying a recursive development which isalso presented in the Appendix.

3. Problem of wavefront slope measurement

In the previous paper (Molodij et al. 1996), we have pre-sented a new tracking method, called granulation tracking,for measuring image motions by which extensive sourcescan control a tiltable mirror in real-time. The principleof this method is to monitor the variation of the imageFourier transform with respect to some reference, in or-der to estimate its motions. The image is detected bytwo detectors (Reticon 128 s photo-diode arrays), whichcover a square field together but which each have reso-lution in one direction only. The reticon has elongated2500 × 25 micron pixels that are sensitive to motion inthe direction of the smaller diode dimension. Therefore,in the granulation image, the cross-section of the two ar-rays defines a square field of view which can be adjustedfrom 2× 2 arcsecond to 12× 12 arcsecond, depending onthe Fried parameter r0 (Rayrole 1992).This tracking method, as in correlation tracking method,uses extended solar granulation images as target struc-ture to detect wavefront aberrations. The size of the ana-lyzed field of view is adjusted so the tracker will success-fully operate under any seeing conditions. One has to keepin mind, however, that wavefront slope measurements aremade to correct pure wavefront tilts. All higher order aber-rations such as coma Zernike polynomials (Z7 or Z8) areinterpreted by the system as image motions because it isnot possible to distinguish between, for example, a tilt inthe wavefront and deformations of the object itself. Thegranulation tracker can not determine the appropriate tiltsof the wavefront surface (which is the best correction interm of Strehl resolution). Such wavefront analyzer sys-tem determine relative displacements of successive imagesin real time. These shifts constitute arbitrary position ofstabilization defined both by the target motions and thetarget structure deformations, which are applied to thetilting mirror.We propose to analyze the problem of the wavefront slopemeasurement calculating the first phase derivative expan-sion on the Zernike polynomials using Primot’s approach(Primot et al. 1990) in order to determine the image qual-ity after image stabilization from wavefront slope mea-surements.The wavefront slope measurements (in x and y direction)can be written:{δφ(x,y)δx

=∑∞j=1 aj

δZjδx

=∑∞j=1 aj

∑∞j′=1 γ

xjj′ Zj′

δφ(x,y)δy

=∑∞j=1 aj

δZjδy

=∑∞j=1 aj

∑∞j′=1 γ

yjj′ Zj′

(16)

where γxjj′ and γyjj′ are the matrices expressed by Noll(Noll 1976) for representing derivatives of Zernike polyno-mials as a linear combination of Zernike polynomials.We can express the wavefront slope measurements overthe circular aperture of the telescope. For instance, the

wavefront slope in x direction, denoted δφδx

is written:

δφ

δx=

∞∑j=1

aj

∞∑j′=1

γxjj′

∫ 2π

0

dθ

π

∫ 1

0

r′ dr′ Zj′(r′, θ). (17)

Let us notice that averaging the Zernike polynomials overthe aperture of the telescope is zero if j′ > 1:{∫

dr W (r) Zj′(r′) = 0∫

dr W (r) Z1(r′) = 1.(18)

So that:

δφ

δx=

∞∑j=1

aj γxj1 =

∞∑l=0

a2l+1,1

√4(l + 1) (19)

where the notation Zj(n = 2l + 1,m = 1) is adopted todefine the concerned Zernike polynomials (the radial de-gree is odd and the azimutal frequency m = 1 becausethe averaging over the circular aperture). The measuredquantity is quite different from pure tilt Zernike polyno-mials. Wavefront tilt Z2 not only appear to define the firstmean phase derivative in x direction, but also the 3rd or-der coma Z8 and higher order aberrations such Z16, Z30,Z46, Z68 etc.. Wavefront tilt Z3 and others Zernike poly-nomial aberrations as Z7, Z17, Z29, Z47, Z69 etc. appearto define the first mean phase derivative in y direction.Therefore, the estimated wavefront reconstructed by thesystem from the measurements can be written:

φ(Rρ) = a δφδx

Z2 + a δφδy

Z3 (20)

with:a δφδx = 12δφδxZ2 =

[12

∑∞l=0

√4(l + 1) a2l+1,1

]Z2

a δφδy

= 12δφδyZ3 =

[12

∑∞l=0

√4(l + 1) a2l+1,1

]Z3.

(21)

Let us remark that effects of anisoplanatism must be takeninto account to understand all the effects that might arisewhen adjusting the analyzed field of view of the targetfrom 2 × 2 arcsecond to 12 × 12 arcsecond. The statisti-cal analysis of angular cross-correlation allows to deter-mine the aberration strenghts of all the higher order aber-rations participating in the modal expansion of the firstphase derivative of Eq. (21) in such situation.Figure 1 shows the normalised angular correlation func-tions of the first polynomials of the expansion concernedfor the 0.9 m telescope diameter (T.H.E.M.I.S.). Thesecorrelations are calculated using the simulated C2

n turbu-lence profile (Fig. 2) from experimental measurements atIzana site by Arcetri University (Barletti et al. 1973). For


Fig. 1. Angular correlation functions for the higher orderaberrations participating in the modal expansion of the firstphase derivative versus the angular separation. The notationZj(n = 2l + 1,m = 1) is adopted to define the concernedZernike polynomials j (n is the radial degree and m the az-imuthal frequency). These correlations are calculated usingthe simulated C2

n turbulence profile (D/r0 = 5) from ex-perimental measurements at Izana site by Arcetri University(Barletti et al. 1973)

a given angular separation α corresponding to the size ofthe field of view analyzed by the granulation tracker, theonly Zernike polynomials defined by Eq. (21) which areuseful in the measured quantity, are the polynomials hav-ing their 50% correlation angle larger than α. The pecu-liar shape of the curves is related to the turbulence profilesimulated, (r0 = 18 cm of Fig. 2) (Barletti et al. 1973).The higher order aberrations participating in the modalexpansion of the first phase derivative of Eq. (21) are cor-related on the field of view delimitated by the granulationtracker so that the measured quantity by the analyzer isvery close to the wavefront slope.

Let us notice that Fig. 1 reveals the strong dependenceof the angular correlations on the angular separation αand the radial degree n. If the decorrelation of the tilts(n = 1) is slow when increasing α, for others polynomialsthe decorrelation is steep. This result is general in themodal analysis of the effects of the atmospheric turbulenceon the wavefronts (Rousset 1993).

Let us now consider how the correction is made by theimage stabilizer system considering the residual wavefronterror Eq. (7) when correcting pure wavefront tilts fromslope measurements.The estimated wavefront tilts by the system are given byEq. (21) so that the residual error becomes:

ε2(α)=∞∑n=1

(n+ 1)<(an,0)2 >+2∞∑l=0

∞∑l′=0

√(l + 1)(l′ + 1)

<a2l+1,1(0)a2l′+1,1(0)>− 4∞∑l=0

√l+1<a2l+1,1(α)a1,1(0)> .

(22)

Fig. 2. C2n turbulence profile from experimental measurements

at Izana site by Arcetri University (Barletti et al. 1973). Theprofiles are integrated from 10 m (above the ground) to 2000 mto give Fried’s parameter r0 at λ = 0.5 µm from 10 cm to22 cm (respectively D/r0 from 9 to 4)

The residual error after image stabilisation expansioncontains angular cross-correlation of Zernike polynomialsj1(n1,m1) and j2(n2,m2): < an1,m1an2,m2 > which canbe calculated introducing minor corrections in Eq. (11):

< aj1(α)aj2(0) >= 3.895√

(n1 + 1)(n2 + 1)

(D

r0

) 53

∫ Latm

0dhC2

n(h) I ′n1+n2,m1+m2(αhR

)∫ Latm

0dhC2

n(h)(23)

where I ′n1+n2,m1+m2(x) is now written:

I ′n1+n2,m1+m2(x)=sn1+n2,m1+m2

√(n1+1)(n2+1)

[K+

1,2∫ ∞0

dKK−14/3Jn1+1(K)Jn2+1(K)Jm1+m2(Kx)+K−1,2∫ ∞0

dKK−14/3Jn1+1(K)Jn2+1(K)J|m1−m2|(Kx)]

(24)

where K+1,2 and K−1,2 ,quantities depending on the coeffi-

cient pairs (j1, j2) parity, are listed in Table 1.The residual wavefront error after image stabilization maybe expressed:

ε2(1, α) = 3.895

(D

r0

) 53

[∞∑n=1

(n+ 1)2In,0(0)


Table 1. Angular cross-correlation Zernike polynomial coeffi-cients

K+1,2 m1 = 0 m1 6= 0& j1 even m1 6= 0& j1 odd

m2 = 0 1√

2 0

m2 6= 0& j2 even√

2 1 0m2 6= 0& j2 odd 0 0 −1

K−1,2

m1 = 0 m1 6= 0& j1 even m1 6= 0& j1 odd

m2 = 0 0 0 0m2 6= 0& j2 even 0 1 0m2 6= 0& j2 odd 0 0 1

Fig. 3. Residual phase variance after compensation versus theangular separation α. For comparison: residual phase variancesafter correction of J = 3 polynomials (N = 1), J = 6(N = 2),J = 10(N = 3) and after correction of pure tilts from wavefrontslope measurements (dark points). Curves are calculated withthe modelised profile at Izana site (D/r0 = 4)

+ 2∞∑l=0

∞∑l′=0

√(l+1)(l′+1)(2l+2)(2l′+2)

×

∫ Latm

0dhC2

n(h) I2l+2l′+4,2(αhR

)∫ Latm

0dhC2

n(h)

− 4∞∑l=0

√(l + 1)2(2l+ 2)

×

∫ Latm

0dhC2

n(h) I2l+2,2(αhR

)∫ Latm

0dhC2

n(h)

]· (25)

In any case the residual phase variance ε2 can be estimatedfrom the knowledge of the Zernike coefficient angular cor-relations.

4. Computation results

The goal of this study is to compare the performance of theimage stabilizer optical system and adaptive optics systemwhich will correct aberrations as tilt, focusing, astigma-tism and coma considering not only the image resolutionbut also the degradation of the compensation by the sys-

tem in the field of view. Note that noise in the wavefrontmeasurement will not be considered and the bandwidth ofthe servo-loop will be considered infinite.

Fig. 4. Isoplanatic domain (twice as isoplanatic angle αiso)corresponding to an image quality criterion of λ

5 , versus wave-length, for the T.H.E.M.I.S. telescope aperture (D = 0.9 m)after compensation of J = 3 first aberrated modes, J = 10modes and after image stabilisation (dark points). Curves arecalculated with the modelised profile at Izana site (D/r0 = 4)

In order to determine the isoplanatic domain surround-ing the target structure on-axis, we have to use a criterionto evaluate the quality of the restored image. This crite-rion is based on the residual phase variance, defined inEq. (15) for the compensation of first aberrated modesand in Eq. (25) for image stabilizer system, which must belower than a given value ε20 (in rad2). An adaptive opticssystem compensating for turbulence disturbances up tothe maximum radial degree N (N = 1 for image stabilizersystem) is characterized by the residual phase variance:

ε2(N,α) =

(D

r0

)5/3

f(N,α) (26)

where f(N,α) is deduced from Eq. (15) in the case of com-pensation of aberrated modes up to the maximum radialdegree N and from Eq. (25) in the case of image stabi-lizer system. The isoplanatic angle αiso (half of the fieldof view) is given by the equality:(D

r0

)5/3

f(N,αiso) = ε20 =

(2π

a

)2

(rad2). (27)

Let us notice that the standard deviation ∆sd =√ε20 may

be expressed in wavelength unit: ∆sd = λa

(rms).In Fig. 3, the residual phase variances after compen-

sation are plotted for T.H.E.M.I.S. telescope (D = 0.9 m)and for the modelised profile at Izana site defined byFig. 2 (r0 = 22 cm). The curves correspond to different


adaptive optics systems compensating for turbulence dis-turbances up to the maximum aberration modes J (i.e.maximum radial degree N) compared to the image stabi-lizer system. These configurations are possible cases whichcould be considered for practical implementation (Kupkeet al. 1994). Such type of curves have been first plotted byChassat in order to manage optimal correction in termsof angular decorrelation by tuning up the number of cor-rected modes.

Fig. 5. Isoplanatic domain (twice as isoplanatic angle αiso)corresponding to an image quality criterion of λ

5, versus wave-

length, for the T.H.E.M.I.S. telescope aperture (D = 0.9 m) af-ter image stabilisation. Curves are calculated with three mod-elised profiles from experimental measurements at Izana siteby Arcetri University (Barletti et al. 1973)

Notice that the C2n profile, chosen to illustrate the

derivations corresponds to a ratio D/r0 = 4. In the threeothers modelised profiles of Fig. 2, the contribution of theturbulence near the ground increases or decreases bothwith the high altitude layers at 4000 m above the telescopealtitude (and probably generated by the Pico del Teide).Therefore, the function f(N,α) in Eq. (26) remains iden-tical when changing the ratio D/r0. In such situations,the residual variance is given by a shift along the y axisof Fig. 3 corresponding to the new value (D/r0)5/3.

First, let us underline the significant reduction of theresidual phase variance on-axis (α = 0) when increasingthe number of corrected modes. The results shown by thecurves (J = 3, 6 and 10) are in perfect agreement with theresidual variance given by Noll (Noll 1976). The imagemotions compensation curve (α = 0) reveals the contri-bution of the error made when correcting pure wavefronttilts from wavefront slope measurements.Secondly, these results allow to understand that for smallfield of view observations, increasing the number of cor-rected mode is the best choice in terms of image quality(i.e. small phase variance) keeping in mind the fast degra-dation of this quality in the field. The obtained residual

phase variance is very small after correction of 10 modes(point at α = 0). On the contrary, the on-axis correctionafter image stabilisation is relatively poor, but the residualphase variance does not vary rapidly with the field angle.At very large angle (α ≥ 10 arcsec), the correction afterimage stabilisation is even better than one obtained aftercorrection of 10 aberrated modes, because of the low wave-front slope angular decorrelation. This demonstrates theimportance of the decorrelation of the higher wavefrontdeformation modes in the field.

Fig. 6. OTF (α = 0) versus the normalised spatial frequency(ρ/R = 2 corresponds to the diffraction limited, i.e. a spatialresolution around 0.145 arcsec at λ = 0.5 µm), for several or-ders of modal corrections (compensation through coma J = 10,compensation of tilts J = 3 and after image stabilization), atD/r0 = 5. The full curve is the diffraction limited compara-tively to the dot curve without correction

Using the relation between the wavelength and theresidual phase variance, Eq. (27) can be written:

λ(µm) =a

2π

√f(αiso) 0.5

(D

r0

)5/6

(28)

where r0 is the Fried parameter calculated at the wave-length λ = 0.5 µm.Figure 4 presents the isoplanatic angle αiso for ε0 = 2π/5(corresponding to λ/5, i.e. a Strehl ratio around 20%) cal-culated by Eq. (28). This value of residual error is some-what arbitrary and rather a poor performance for an adap-tive optics system but can be sufficient for many astro-nomical observations. As expected, the isoplanatic angleαiso increases with the increase of the observation wave-length. Note that αiso after image stabilisation is largerthan αiso after compensation of 10 first aberrated modeswhen observing at wavelength larger than 0.7 µm. Let usremind that for any field point at an angular distance tothe target structure on-axis smaller or equal to αiso, theresidual error is lower than 2π/5. Depending on the choice


of a maximum residual phase, the considered observationwavelength domain is not totally accessible (bandwidthfrom 0.5 µm up to 0.9 µm with T.H.E.M.I.S.). This isshown in Fig. 5 which present the isoplanatic domains af-ter image stabilisation for three different values of Friedparameter r0 obtained by the profile models (Fig. 2). Forinstance, with a Fried parameter r0 = 13 cm, the obser-vation wavelength must be larger than λlimit = 0.78 µmin order to achieve a residual error better than λ/5. Forr0 = 22 cm, λlimit is 0.5 µm.

5. Optical transfer functions

The image quality obtained, after the T.H.E.M.I.S. adap-tive optics system is applied, can be evaluated by compar-ing the restaured image frequencies of the optical tranferfunction in the field of view. The optical transfer functionafter correction (OTF ) can be defined as the mathemat-ical expectation of the cross-correlation function of thecomplex amplitude ψ(r) of the compensated wavefront.Assuming that scintillating effects are negligible, this am-plitude can be written:

ψ(r) = exp{iφc(r)} (29)

where the wavefront after compensation φc(r) is the dif-ference between the wavefront φ(r) coming from the ob-

served source and the estimated wavefront φ(r) recon-structed by the adaptive optics system from the measure-ment at angular distance α.Assuming the near field approximation is verified, theOTF can be written:

OTF (ρ, α)=

∫dr< ψ

(r+

ρ

2

)ψ∗(r−

ρ

2

)>W

(r+

ρ

2

). W

(r −

ρ

2

)(30)

where W (ρ) is the normalised function defined by Eq. (3).Having remarked that phase is a Gaussian random func-tion (Roddier 1981), Eq. (30) becomes:

OTF (ρ, α)=

∫dr exp

(−

1

2<[φc

(r+

ρ

2

)−φc

(r−

ρ

2

)]2>

)

. W(r +

ρ

2

)W(r −

ρ

2

). (31)

The statistics of the Zernike coefficients for an expansionof Kolmogorov turbulence phase distortion have been ap-plied by Wang and Markey to calculate the OTF for apoint source case (α = 0) (Wang & Markey 1978). Wehave extended the method to cases when the observingsource is extended in order to evaluate the degradation inthe field of view of the image due to anisoplanatism afteradaptive optics compensation.Using Eqs. (1) and (6) for the wavefront expansions, the

argument of the exponential of Eq. (31) can be expressedas:

E(r,ρ) = −1

2<[φ(r+

ρ

2

)−φ

(r−

ρ

2

)]2>

−1

2<[φ(r+

ρ

2

)−φ

(r−

ρ

2

)]2>

+ <[φ(r+

ρ

2

)−φ(r−

ρ

2

)][φ(r+

ρ

2

)−φ

(r−

ρ

2

)]> . (32)

We recognize in the right half of the equation the first termas the wave structure function. For a Kolmogorov type ofturbulence, the wave structure function for a plane wavecan be expressed:

<[φ(r+

ρ

2

)−φ

(r−

ρ

2

)]2>=D(ρ)=6, 88

(ρ

r0

)5/3

.(33)

The cross-correlations of the Zernike coefficients < aiaj >(α) are calculated when the reference source target isat angular distance α from the observed source forJ corrected modes by the adaptive optics system (seeAppendix). The turbulence profiles C2

n used in the cal-culation of the angular cross-correlation are presented inFig. 2:

<[φ(r+

ρ

2

)−φ(r−

ρ

2

)]2>=

J∑i=2

J∑j=2

< aiaj > Hi,j(r,ρ)

(34)

<[φ(r+

ρ

2

)−φ

(r−

ρ

2

)] [φ(r+

ρ

2

)−φ

(r−

ρ

2

)]>=

∞∑i=2

J∑j=2

< aiaj > (α) Hi,j(r,ρ). (35)

Introducing the notation:

Hi,j(r,ρ)=[Zi

(r+

ρ

2

)−Zi

(r−

ρ

2

)] [Zj

(r +

ρ

2

)− Zj

(r −

ρ

2

)]. (36)

The OTF in the field of view is obtained evaluating thefollowing numerical expression as indicated by Wang &Markey (Wang & Markey 1978):

OTF (ρ, α) =4

π

∫ π/2

0

dθ

∫ rmax

0

dr r exp[E(r,ρ)] (37)

with

E(r,ρ)'−1

2D(ρ)+

J∑i=2

J∑j=2

[< aiaj > (α)−

1

2< aiaj >

]


Hi,j(r,ρ)+

Jlimit∑i=J+1

J∑j=2

< aiaj > (α)Hi,j(r,ρ) (38)

and

rmax = −ρ

2cos(θ − ϕ) +

√1− (

ρ

2sin(θ − ϕ))2

where ϕ is the argument of ρ.

First, we have verified that the infinite sum can bestopped at Jlimit = 91 when the compensating phase dis-tribution is an expansion through coma J = 10 and α = 0,as indicated by Wang and Markey.Secondly, when α ≥ 10 arcsec with the considered turbu-lence profile, the cross-correlation terms can be approxi-mated by truncating the infinite sum over a more appro-priate limit around Jlimit = 40 (for an estimated conver-gence at 10−8). This is due to the fact that the loss ofcross-correlations of higher order aberrations is more im-portant in the field of view (depending on α) than thelower orders.

Figure 6 shows the OTF (α=0) versus the spatial fre-quency calculated with the turbulence profile presented inFig. 2 (r0 = 18 cm) and zero field of view for 10 ordersof correction (compensation through coma) and for 3 or-ders of correction (compensation of tilts) comparativelyto image motion correction (image stabilization). The re-sults obtained for J = 10 and J = 3 agree exactely withthe analytical calculation reported by Wang and Markey.An analytic expression of theOTF taking into account the

Fig. 7. OTF (α=0) versus the normalised spatial frequencyρ/R, for various level of turbulence taking into account the cen-tral obscuration of telescope T.H.E.M.I.S. (u = d/D = 0.44).The C2

n turbulence profiles used in the calculations, have beensimulated from experimental measurements made at Izana site.The full curve is the diffraction limited

central obscuration of the telescope has been published byPerrier (Perrier 1989). Therefore, the OTF can be written:

OTFco(ρ, α) = OTF (ρ, α) .G(ρ, u)

OTFdiffrac(39)

where u = dD

is the linear central obscuration by compari-son to the T.H.E.M.I.S. diameter D. In this case u = 0.44.Figure 7 presents the image quality improvement aftercompensation by the image stabilizer system taking intoaccount the central obscuration of the magnetograph forvarious levels of turbulence and zero field of view. One ob-serves that the image quality improvement grows when theturbulence strength expressed by Fried’s parameter r0 in-creases, i.e., when the ratio D/r0 decreases. For instance,in Fig. 7 the image motion stabilization is insufficient forhigh turbulence levels caracterized by a ratio D/r0 ≥ 6when imaging in the visible wavelength (λ = 0.5 µm).High spatial frequencies are not restored by image stabi-lizer system compensation.

Effects of anisoplanatism are presented in Fig. 8. TheOTF for compensation through the coma J = 10 areshown for different angular separations in the field ofview for the same turbulence profile. As expected compar-ing with Fig. 4, one observes that the image quality im-provement when compensating 10 modes decreases rapidlywhen the angular separation increases for an extendedsource. At very large angle (α ≥ 20 arcsec) in Fig. 8,the image quality after compensation of 10 modes forD/r0 = 5 is hardly better than the one obtained withoutcorrection, because of the wavefront angular decorrelation.

Fig. 8. OTF (α) versus the normalised spatial frequency ρ/R,for several angular separation α for a compensation by theadaptive optics system through the coma J = 10. The fullcurve is the diffraction limited

In Fig. 9 the relative OTF corresponding to the ra-tio of the OTF after compensation through J = 10 bythe OTF after compensation through J = 3, OTF =


Fig. 9. Relative OTF = OTFJ=10 / OTFJ=3 versus the nor-malised spatial frequency (ρ/R) for several angular separationsα in the field of view. Curves are calculated with the modelisedprofile (r0 = 18 cm from experimental measurements at Izanasite by Arcetri University (Barletti et al. 1973)

OTFJ=10 / OTFJ=3, are plotted versus the normalisedspatial frequency for different angular separations α.Figure 9 allows to understand the problem of the degrada-tion of the compensation in the field of view due to aniso-planatism when increasing the order of correction fromJ = 3 up to J = 10. In the case of point source correction(curve α = 0), increasing the order of correction permitsa higher-level image quality. In the case of an extendedsource (α ≥ 5 arcsec for the turbulence profile studied),increasing the order of correction can be detrimental tothe image quality. For instance, the curve α = 5 arcsec ofFig. 9 shows that higher spatial frequencies (ρ/R ≥ 1.4)are degraded when the compensating phase distribution isan expansion through coma comparatively to a tilts com-pensation by the adaptive optics system. This anisopla-natism effect can be understood by remarking that theloss of correlation is faster for the higher order coefficientsthan for low order. Of course, this behaviour is empha-sized in the comparison between OTFJ=10 and OTFJ=3

because of the large isoplanatic patch for the tilt aber-ration. Similar conclusions were found by Abitbolt andBen-Yosef (Abitbolt & Ben-Yosef 1991).

We present the results of numerical simulations to ana-lyze the effects of anisoplanatism when observing extendedsources with possible configurations of adaptive optics sys-tem which could be considered for practical implementa-tion on T.H.E.M.I.S. We compute analytically the long ex-posure optical tranfer functions OTF in the field of view(for different angular separation α) in order to simulatethe filtering process of the atmospheric turbulence on thespatial frequencies of a long exposure image of the solargranulation.

Figure 10 demonstrates the necessity to use adaptiveoptics system to compensate the terrestrial atmospheric

perturbations on wavefronts coming from the Sun to thetelescope (comparison between Fig. 10a and Fig. 10b).Figure 10a shows a 15 arcsec2 field of view of the solargranulation obtained with short exposures (' few ms inimagery mode). This image allows to simulate the effectsof atmospheric turbulence in cases when long exposure areindispensable for accurate measurements in spectroscopymode (' 300 ms) before and after compensation. A com-parison of the effects of anisoplanatism is shown on anextended image of solar granulation after real time com-pensation by the image stabilizer system and by a 10controlable modes adaptive optics system (Fig. 10c andFig. 10a).Let us underline the significant improvement of the im-age quality when compensating the 10 first aberrationmodes on this small field of view of the image granula-tion (Fig. 10d). By comparison, the on-axis correction isrelatively good with the image stabilizer optical system.It reveals the importance of the contribution of the tiltsin the adaptive optics compensation when observing witha 90 cm telescope aperture for mean atmospheric turbu-lences. Notice also that tilt variances represent 90% of thetotal variance phase (Noll 1976).Figure 11 presents the same comparison on a very largefield of view (around 50 arcsec2). One observes the sig-nificant increase of anisoplanatism effects in the fieldof view with the increase of the number of correctedmodes (Fig. 11c: image stabilization with a tip-tilt mir-ror, Fig. 11d: compensation through the coma J = 10).This simulation allows to understand that for small fieldof view observations, compensation of the 10 first aber-ration modes is the best choice in terms of image qualitykeeping in mind the fast degradation of this quality in thefield of view as shown by Fig. 11d. In contrary, with imagestabilization, the image quality does not vary fastly withthe field angle. It reveals the important contribution ofthe large isoplanatic patch of the tilt aberrations for theadaptive optics compensation. At very large angle (α ≥20 arcsec), the correction with the image stabilizer opti-cal system (Fig. 11c) is even better than the one obtainedwith the compensation of 10 first aberration modes byadaptive optics system (Fig. 11d).

Figure 12a demonstrates the capacity of image sta-bilizer system for large field of view observations andmedium image quality. The on-axis correction is relativelygood considering the evaluation of the Fried’s resolution(0.45 arcsec in the center part of the field of view) andthe degradation does not vary fastly with the field angle.This is due to the dominant contribution of the lawest al-titude layers in the angular correlation of the wavefronts.This system is well designed for solar observations likesunspots (up to 30 arcsec), active regions and protuber-ances (few arcminutes). Notice that the temporal analysisof solar phenomena needs also an homogeneous field ofview.


Fig. 10. A) Solar granulation at diffraction (short exposure). Image obtained by R. Muller. The size of the field-of-view is15 × 15 arcsec. B) Effects of the turbulence for long exposure. The Fried’s resolution is around 0.8 arcsec (r0 = 18 cm). Theturbulence profile has been simulated from experimental measurements at Izana site by Arcetri university (Barletti et al. 1973).C) Anisoplanatism effects after real-time compensation by the image stabilizer optical system. D) Anisoplanatism effects aftercorrection of 10 controlable modes (tilt, focusing, astigmatism and coma)


Fig. 11. A) Solar granulation at diffraction (short exposure). Image obtained by R. Muller. The size of the field-of-view is50x50 arcsec. B) Effects of the turbulence for long exposure. The Fried’s resolution is around 0.8 arcsec (r0 = 18 cm). Theturbulence profile has been simulated from experimental measurements at Izana site by Arcetri university (Barletti et al. 1973).C) Anisoplanatism effects after real-time compensation by the image stabilizer optical system. D) Anisoplanatism effects aftercorrection of 10 controlable modes (tilt, focusing, astigmatism and coma)


Fig. 12. Evaluation of Fried resolution in the field-of-view (50 arcsec) for image stabilizer system (top: A) and (bottom: B)for adaptive optics system after correction of 10 controlable modes (tilt, focusing, astigmatism and coma). By comparison, theresolution without correction is only 0.8 arcsec (r0 = 18 cm)


By comparison, Fig. 12b allows to understand that forsmall field of view observations, the 10 corrected modesadaptive optics system is the best choice in terms ofimage quality keeping in mind the fast degradation ofthe quality in the field (resolution of 0.25 arcsec up to0.62 arcsec in the 50 arcsec field). This demonstrates theimportance of the decorrelation of the higher wavefrontdeformation modes in the field. So, the fast degradationof the resolution in the field will be problematic for manyastronomical observations. Nevertheless, this system willbe well designed to the observation of concentrated struc-tures of the magnetic field (0.2 arcsec), penumbra fibrils(0.5 arcsec) or details of sunspots.

6. Conclusion

The generalized theoretical analysis of correlation func-tions leads to an accurate modelling of anisoplanatism.The residual wavefront error derivation presented inSect. 2 and 4 provides a useful analysis to evaluate theeffects of anisoplanatism upon the performance of solaradaptive optics systems. The theoretical analysis uses thewavefront expansion on the Zernike polynomials which isalso used in Sect. 3 to analyze the problem of the wave-front slope measurements on the granulation image whencompensating with a tiltable mirror. A detailed evalua-tion of the anisoplanatism is deduced from the Zernikepolynomial angular correlations, derived using the Mellintransform technique. In Sect. 5, the optical transfer func-tions are evaluated for a Zernike expansion following theWang and Markey approach. The calculation of OTF tocases when the observing source is extended is developedin order to evaluate the degradation in the field of viewof the image due to anisoplanatism after adaptive opticscompensation.We compute the isoplanatic angle corresponding to aλ/5 maximum wavefront error in the field of view forT.H.E.M.I.S. after compensation by the image stabi-lizer optical system and different observation wavelengths.Depending on the level of the strength of the turbulence,the accessible wavelength range is limited. We compare theperformances of the actual image stabilizer optical systemand a project of 10 controlable modes of correction (tilt,focusing, astigmatism and coma). We demonstrate the ca-pacity of 10 controlable corrected modes for high qualitycompensation but to the price of a small field of view whilethe capacity of the image stabilizer optical system is formedium quality compensation and large field of view ob-servations.The study may be useful for the design of future adaptiveoptics system for solar observations. Depending of the spe-cific goals, large or small field of view and medium or highimage quality, the system may be totally different.

The authors are grateful to R. Muller for granulationimages of the sun from the Pic-du-midi Observatory.

A. Appendix

As demonstrated by Sasiela (Sasiela 1994), the proper-ties of Mellin transforms (see for instance (Colombo 1959;Dautray & Lions 1987)) and Gamma functions are use-ful in problems dealing with wave propagation in turbu-lence. This technique is applied here to solve the integralin Eq. (12), Eq. (24), Eq. (34) and Eq. (35).The Mellin transform pair is given by:

F (s) =

∫ ∞0

dKKs−1 f(K) and

f(K) =1

2iπ

∫ σ+∞

σ−∞dsK−s F (s). (A1)

The general form of the integral to solve has been pre-sented by Chassat (Chassat 1989) and can be written(η = 14

3 ):

I(x) =

∫ ∞0

dK K−η Jα(K) Jγ(K) Jβ(xK). (A2)

From the properties of the Mellin transforms (Colombo1959; Dautray & Lions 1987) and using Fn(s) =∫∞

0 dKKs−1 Jn(K), the integral becomes:

I(x) =1

(2iπ)2

∫ c+∞

c−∞

∫ c+∞

c−∞Fα(s)Fβ(t)x−tFγ

× (−η − t+ 1− s)dsdt. (A3)

Tables of Mellin transforms (Sasiela 1994; Colombo 1959;Dautray & Lions 1987) are helpful to solve Eq. (A3).Moreover, the Mellin transform of functions can usuallybe expressed as the ratio of Gamma functions. We use thenotation:

Γ

[x1, x2, ...., xny1, y2, ...., ym

]=

Γ(x1) Γ(x2) ....Γ(xn)

Γ(y1) Γ(y2) ....Γ(ym)· (A4)

We have for Jn the (n)th order Bessel function of the firstkind:

Fn(s) = 2s−1 Γ

[s2 + n

2n2 −

s2 + 1

]

with Re(−n) < Re(s) <3

2· (A5)

It leads to the Mellin-Barnes integral:

I(x) =1

(2iπ)2

∫ +i∞

−i∞

∫ +i∞

−i∞2−ηx−2t

Γ

−s− t+ α−η+12 , t+ β

2 , s+ γ2

1 + t+ s+ α+η−12 , 1− t+ β

2 , 1− s+ γ2

dsdt. (A6)

This integration can be performed using the method ofpole residues. The value of the integral, as given by


Cauchy’s formula, is just 2iπ times the sum of the residuesat the enclosed poles. The result can be expressed in termsof generalised hypergeometric functions (Abramovitz &Stegun 1965). For instance, in the case x = αh

R> 1, one

of the interesting cases, the integral is given by:

I(x) = F4

[1

2(α+ γ + β −

11

3),

1

2(α+ γ − β −

11

3), α+ 1,

γ + 1;1

x2,

1

x2

]Γ[1

2 (α+ δ + β − 113 ]

Γ[γ + 1]

Γ[12 (α+ γ − β − 11

3 ]

Γ[α+ 1]

(−1)2p+2q

−π2−

113 x−ni−nj+

53 sin

(π

2(α+ γ − β −

17

3)

)(A7)

where:

F4(α, β, γ, γ′;x, y) =∞∑m=0

∞∑n=0

(α)m+n (β)m+n xm yn

(γ)m(γ′)nm!n!

with the notation (a)n =Γ(a+ n)

Γ(a)· (A8)

When the function is expressed in terms of generalisedhypergeometric functions, the integral leads to the morerestrictive convergence condition which can be expressedas: αh

R< 2. This condition determines the maximum field

of view which can be reached by this method.The limited convergence domain encountered in the eval-uation of integrals involving the product of three Besselfunctions has been first remarked by Tyler (Tyler 1990).Chassat proposed a method using the Bessel recurrencylaw (Chassat 1992):

Jµ+2(K) =2(µ+ 1)

KJµ+1(K)− Jµ(K). (A9)

Two easily evaluable Mellin integrals appear, I1(x, µ, β, η)and I2(x, µ, β, η)

I1(x, µ, β, η) =

∫ ∞0

dKK−η J2µ(K)Jβ(Kx) (A10)

I2(x, µ, β, η)=

∫ ∞0

dKK−η Jµ(K)Jµ+1(K)Jβ(Kx).(A11)

With the notation µ = inf(α, γ) and ε = |α − γ|,Eq. (A2) becomes:

I(x) =

∫ ∞0

dK K−η Jµ Jµ+ε(K) Jβ(xK)

= I(µ, ε, η, β)(x). (A12)

The Bessel reccurency law becomes reccurency law be-tween Mellin integrals which can be written as:

I(µ, ε− 1, η, β) = 2(µ+ ε− 1)I(µ, ε, η + 1, β)−

I(µ, ε− 2, η, β). (A13)

Now, we can use the Mellin transform tables to solveI(x, µ, β, η) and I(x, µ, β, η) which can be expressed asMellin-Barnes integrals of the following type:

1

2iπ

∫ c+i∞

c−i∞G(t)F (1− t− η)xt+η−1 dt (A14)

where G and F are the Mellin transform of g and f re-spectively. g and f are defined as:{f(aK) = Jβ(Kx)g(K) = J2

µ(K)to solve: I1(x, µ, β, η)(A15)

{f(aK) = Jβ(Kx)g(K) = Jµ(K)Jµ+1(K)

to solve I2(x, µ, β, η).(A16)

We have the Mellin transforms:

g(K)=J2µ(K) −→ G(s) =

1

2√π

Γ

[s2 +µ, 1−s

2µ+1− s

2 , 1− s2

]with Re(−2µ) < Re(s) < 1 (A17)

g(K) = Jµ(K)Jµ+1(K) −→ G(s)

=1

2√π

Γ

[s+1

2 + µ , 1− s2

µ+ 3−s2 , 3−s

2

]with Re(−2µ− 1) < Re(s) < 2. (A18)

The Mellin transform of f(Kx) is given by Eq. (A2).Finally, using the Cauchy’s formula, we find:If x ≤ 2

I1 (x, µ, β, η) =∞∑p=0

1

2√π

(−1)p

p!

(x2

)2p+η

Γ

[12 + µ+ p , β2 − p−

η2

−p+ µ+ 12 ,

12 + p , p+ β

2 + 1 + η2

]+

∞∑p=0

1

2√π

(−1)p

p!

(x2

)2p+β

Γ

[ 1−η+β2 + µ+ p , −p+ η+β

2

µ− p− β+η+12 , 1+η−β

2 − p , p+ β + 1

](A19)

I2 (x, µ, β, η) =∞∑p=0

1

2√π

(−1)p

p!

(x2

)2p+η+1

Γ

[32 + µ+ p , −p− η+1−β

2

µ− p+ 12 ,

12 − p , p+ η+3+β

2

]+

∞∑p=0

1

2√π

(−1)p

p!

(x2

)2p+β

Γ

[1−η+β

2 + β + p, −p+ η+1+β2

µ− p+ 1 + η−β2 , 1 + η−β

2 − p, p+ β + 1

](A20)


and x > 2

I1 (x, µ, β, η) =∞∑p=0

1

2√π

(−1)p

p!

(x2

)−2p−2µ+η−1

Γ

[12 + µ+ p , p+ µ+ 1−η+β

2

p+ 2µ+ 1 , p+ µ+ 1 , −p− µ+ η+1+β2

](A21)

I2 (x, µ, β, η) =∞∑p=0

1

2√π

(−1)p

p!

(x2

)−2p−2µ+η−2

Γ

[32 + µ+ p , p+ µ+ 1− x+β

2

p+ 2µ+ 2 , p+ µ+ 2 , −p+ µ+ x+β2

]. (A22)

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