Wavefront propagation in turbulence: an unifiedapproach to the derivation of angular
correlation functions
Guillaume Molodij
LESIA-Observatoire de Paris-Meudon, CNRS, associé à l’Université Pierre et Marie Curie-Paris 06 et à l’Université ParisDiderot-Paris 07, Paris, France ([email protected])
Received March 16, 2011; revised June 26, 2011; accepted June 27, 2011;posted July 7, 2011 (Doc. ID 144355); published July 28, 2011
1. INTRODUCTIONSeveral authors have presented studies on the correlation ofphysical quantities related to the optical phase to determinethe effects of the terrestrial atmosphere along the wave pro-pagation path to the observer [1,2]. In many practical applica-tions of adaptive optics (AO), phase corrections that are basedon turbulence information for one propagation path must beapplied to propagation on a second, slightly different path.Knowledge of the correlation functions, suitably integratedover the various layers of turbulence, is necessary to optimallyestimate the phase and are applicable to a number of physicalsituations. If the phase fluctuations produced by the two pathsare highly correlated there is no significant difference be-tween the optimum phase correction for the two paths, andthey are said to lie in the same isoplanatic patch. On the otherhand, as the two paths are separated by many isoplanaticpatches, their phase fluctuations become completely indepen-dent and adaptive phase correction can actually degrade sys-tem performance.
The derivation of the correlation functions are applicable toa number of physical situations. This commonly used expres-sion for the wave structure and mutual coherence function foran optical wave propagating in a turbulent atmosphere intro-ducing the effects of the finite scale have been investigated byseveral authors [3–10]. These results are needed to correctpropagation code calculations in which an effective innerscale must be introduced due to mesh point separationsand a finite outer scale because of total mesh limitations. Forinstance, the effect of a finite spatial coherence outer scale onthe covariances of angle-of-arrival fluctuations has been pre-sented to discuss the influence of the model used to describethe atmospheric turbulence [11].
In order to investigate isoplanatism of phase corrections, itis useful to expand the phase in an orthogonal set of polyno-mials and calculate the spatial correlation functions of the in-
dividual expansion coefficients [12,13]. Zernike polynomialshave been used to describe the modal behavior of an AOsystem when compensating the turbulence [14]. Derivationof the modal correlation functions are useful for near-ground propagation where the outer scale becomes smalland for upper altitude propagation where the inner scale be-comes large. Outer scales have been investigated to find thatthe effects on large telescopes are dependent more on themagnitude of the outer scale than on the shape of the out-er-scale vertical profile [7].
Typical examples of derivation of correlation functions oc-cur in laser beams propagating through atmospheric turbu-lence, in ground-to-satellite communication where adaptivephase corrections based on one link are applied to the other,or when imaging of extended sources is attempted from theground through high altitude turbulence [15,16]. Correctionsfor a laser beam propagating through atmospheric turbulenceare given for turbulence with finite inner and outer scales thathave been established and are useful for determining lengthscales for wave optics propagation codes, since such codesintroduce artificial outer and inner scales due to finite gridsize and mesh-point separation [5].
In this paper, the formalism to generalize the derivation ofangular correlation functions is presented. The problem canbe reduced to the evaluation of an integral over the magnitudeof the spatial transform coordinate over the wave propagationpath is shown. The expression contains extensible aperturefilter functions to take into account the spherical propagation,filter, and analytical functions to determine piston, tilt, andhigher-order aberrations on an aperture, considering the finitesize receivers or sources. For all problems that have standardforms, such as the inner- and outer-scale effects, one can de-rive analytical solutions.
The process of setting up problems of wave propagationthrough turbulence and reducing the expressions to integrals
1732 J. Opt. Soc. Am. A / Vol. 28, No. 8 / August 2011 G. Molodij
is lengthy [17]. The integrand of the integral consists of theproduct of functions of hypergeometric types (a hypergeo-metric function multiplied by a power of the variable). Theintegral over the spatial transform coordinate can be per-formed with Mellin transform techniques so that the solutiontakes the form of a generalized hypergeometric function,which is expressible as a series that converges rapidly formany cases of interest that pertain to atmospheric turbulence.After performing the integration, the problem is then reducedto an integration along the propagation direction and can beevaluated analytically when using the Hufnagel–Valley modelof turbulence [18]. This approach has proved extremely usefulin many applied physics problems, including the analysis ofelectromagnetic propagation in a turbulent medium [19,20].
Section 2 presents a generalization of the angular correla-tion derivations of quantities related to the phase fluctuations.Section 3 provides the expressions of the operators and filterfunctions to determine physical quantities of interest such asthe angle-of-arrivals, the higher-order aberrations, consideringthe finite size of the receivers or the sources, and the inner-or the outer-scale effects. Section 4 shows two applications ofthe method: first to quickly evaluate the performance of an AOsystem, and second to analyze the effect of the outer-scaleeffect of the turbulence on the correlation of the first andsecond derivatives by comparison to the Zernike modes.
2. GENERALIZATION OF THE ANGULARCORRELATION DERIVATIONLet GðRi~ρÞ be the product of the phase wave φiðRi~ρÞ andspatial function Mð~ρÞ. g is the measurement over a circularaperture of radius Ri from a source i:
gi ¼Z
d2~ρWð~ρÞφiðRi~ρÞMð~ρÞ; ð1Þ
where Wð~ρÞ is the pupil filtering function of the normalizedvariable ~ρ:
Wð~ρÞ ¼�
1π : if j~ρj ≤ 10 : else where
: ð2Þ
Invoking the stationarity properties of the atmospheric tur-bulence [1], the angular correlation of the quantity G comingfrom two distinct sources separated by angle α becomes
Let Bφ½Rð ~ρ1 − ~ρ2Þ; α� ¼ hφ1ðR ~ρ1Þφ�2ðR ~ρ2Þi, the phase covar-
iance at the ground, assuming the small perturbation hypoth-esis, the near field, and the statistical independence of theatmospheric layers assumptions [21]
Bφ½Rð ~ρ1 − ~ρ2Þ; α� ¼Xlayersl
Bφl½αhl~iþ R1ðhlÞ ~ρ1 − R2ðhlÞ ~ρ2�; ð6Þ
where Bφl is the phase covariance of the turbulent layer l. ~i isthe unit vector determined by the direction of the two sources,as indicated in Fig. 1, and R1ðhlÞ and R2ðhlÞ depend on thealtitude of the layer, with R1ðh ¼ 0Þ ¼ R2ðh ¼ 0Þ ¼ R closeto the observer.
Introducing the Dirac function δ½ ~κ2 − ζðhlÞ ~κ1�,
δ½ ~κ2 − ζðhlÞ ~κ1� ¼Z
d2~ρ exp½2iπ~ρð ~κ2 − ζðhlÞ ~κ1Þ�; ð9Þ
hg1g�2iðαÞ ¼Xlayersl
Zd2 ~κ1 ~Ml1ð ~κ1Þ ~M�
l2½ζðhlÞ ~κ1�
× exp½−2iπ ~κ1ð ~ρ1 − ζðhlÞ ~ρ2Þ�
×Z
d2~ηðhlÞ exp½−2iπ ~κ1~ηðhlÞ�Bφl½αhl~i
þ R1ðhlÞ~ηðhlÞ�: ð10Þ
Fig. 1. Geometry of the propagation problem.
G. Molodij Vol. 28, No. 8 / August 2011 / J. Opt. Soc. Am. A 1733
The integral over ~η is the Fourier transform of the covari-ance function of the phase Bφ. Assuming a fully developedKolmogorov turbulence [22], the power spectrum of the phasededuced from the Von–Karman model is
Wφðj ~K jÞ ¼ k2
cosΩ 0:033ð2πÞ−23ðK2
þ K20Þ−
116 exp½−ðK=k0Þ2�C2
nðhÞdh; ð11ÞwithK0 ¼ 1
2πL0and k0 ¼ 1
2πl0. L0 and l0 are the limit scales of theturbulence, k is the wave number, and Ω is the zenital angle.
Invoking the Wiener–Khinchine theorem, the discrete sumon the layers of the turbulence leads to a continuous integralover the atmosphere depth Latm:
hg1g�2iðαÞ ¼0:033ð2πÞ−2=3k2
cosΩ
ZLatm
0dh
C2nðhÞ
R21ðhÞ
Zd2~κ ~Mð~κÞ ~M�½ζðhÞ~κ�
× ½ð~κÞ2þð2πL0Þ−2�−116 exp½−ðj~κj2πl0Þ2�
×exp
�2iπαh~κ~iR1ðhÞ
�exp½−2iπ~κð ~ρ1 − ζðhÞ ~ρ2Þ�: ð12Þ
The function ζðhÞ characterizes the relative geometric propa-gation between the two waves coming from the two sources(e.g., ζðhÞ ¼ 1 for the two plane waves case). The parameter αis the angular separation between the source of reference andthe source of interest. Lo and lo are the outer and the innerscale of the turbulence. The function of the frequency cor-responds to the different analytical operators applied anddetailed in the next section.
3. OPERATORS AND FILTER FUNCTIONSThe filter function of the phase spectrum of φðR~ρÞ is applied toderive the phase over a circular aperture. In the Fourier space,the corresponding spectral function is real with an analyticalexpression. In the Noll formalism [14], phase measurement ona telescope aperture of radius R is defined by the implicitfunction Wð~ρÞ of the Fourier transform [Eq. (4)]. Anothernotation of the pupil filtering function (piston mode) in polarcoordinates (κ, ϕ) is
~Mð~κÞ ¼ 2J1ð2RπκÞ2Rπκ : ð13Þ
The phase covariance is given in Eq. (12), withGðR~ρÞ ¼ φðR~ρÞ.
Applied on the phase, an analytical operator defines a spa-tial operator and an analytical spectral function in the Fourierspace. The first phase derivative is, for instance,
MðR~ρÞ ¼ δφðR~ρÞδx ; and ~MðκÞ ¼ 2iπκx: ð14Þ
The angle-of-arrival is defined as the mean first derivative ofthe phase over the pupil. GðR~ρÞ is the product of the phasederivative and the pupil filter function
GðR~ρÞ ¼ Wð~ρÞ δφðR~ρÞδx : ð15Þ
Considering the derivative along the axis ~x, the Fouriertransform of the function MðR~ρÞ is written in polarcoordinates (κ, ϕ)
Mðκ;ϕÞ ¼ 2i cosðϕÞJ1ð2πκÞ; ð16Þ
and along the perpendicular axis ~y:
Mðκ;ϕÞ ¼ 2i sinðϕÞJ1ð2πκÞ: ð17Þ
The wavefront curvature gives the measurement of the meanphase Laplacian over the pupil [23]
GðR~ρÞ ¼ Wð~ρÞ� δ2δx2 φðR~ρÞ þ
δ2δy2 φðR~ρÞ
�; ð18Þ
to obtain ~MðκÞ ¼ −4πκJ1ð2πκÞ.In the Zernike polynomial decomposition, MðR~ρÞ defines
the Zernike coefficients to obtain the derivation of the angularmodal correlations [13].
GðR~ρÞ ¼ Wð~ρÞZjð~ρÞφðR~ρÞ; ð19Þ
where Zjð~ρÞ are defined in polar coordinates (ρ, θ) by aproduct of functions ρ and functions θ [14]:
Zjðρ;θÞ ¼ffiffiffiffiffiffiffiffiffiffiffinþ 1
p8<:
Rmn ðρÞ
ffiffiffi2
pcosðmθÞ : j even and m ≠ 0
Rmn ðρÞ
ffiffiffi2
psinðmθÞ : j odd and m ≠ 0
R0nðρÞ : m¼ 0
;
ð20Þ
with; Rmn ðρÞ ¼
Xn−m2s¼0
ð−1Þsðn − sÞ!s!½nþm
2 − s�!½n−m2 − s�! ρn−2s:
The Fourier transform ~MðKÞ ¼ Qjðκ;ϕÞ is [14]
Qjðκ;ϕÞ¼ffiffiffiffiffiffiffiffiffiffiffinþ1
p Jnþ1ð2πκÞπκ
×
8>><>>:ð−1Þn−m2 im
ffiffiffi2
pcosðmϕÞ : j even and m ≠ 0
ð−1Þn−m2 imffiffiffi2
psinðmϕÞ : j odd and m ≠ 0
ð−1Þn2 : m¼ 0
; ð21Þ
where JnðxÞ is the Bessel function of the nth order and i is theimaginary unit.
GðR~ρÞ can be expressed as a differential function of aquantity related to the phase by the product
where MðR~ρÞ is related to the physical quantity of interest.The Fourier transform of the differential operator is shownin Eq. (12) and is given by exp½−2iπ~κð ~ρ1 − ~ρ2Þ�.
Two examples:
1. the difference of pistons in interferometry
GðR~ρÞ ¼�δ�~ρ −
~ρ02
�− δ
�~ρþ
~ρ02
��Wð~ρÞφðR~ρÞ: ð23Þ
2. The seeing monitor
GðR~ρÞ ¼�δ�~ρ −
~ρ02
�− δ
�~ρþ
~ρ02
��Wð~ρÞ δφðR~ρÞδx : ð24Þ
1734 J. Opt. Soc. Am. A / Vol. 28, No. 8 / August 2011 G. Molodij
Following the analogy with the differential operator, the an-gular operator GðαÞ, where α is the separation angle betweentwo directions of observation is
~MðαÞ ¼ exp½ð2iπαh ~κ · iÞ�: ð25Þ
4. APPLICATIONSA. Simple Derivation of the Isoplanatic Field of Viewafter AO CompensationA simple expression of the isoplanatic field of view after AOcompensation is derived to the purpose of evaluating the AOsystem limitations in regard to the scientific requirements ex-pected for astronomy. Considering the perfect compensationcase of an AO system of J liberty degrees, the object wave-front ~ϕobs is corrected by the phase reconstruction at theJth order of the reference wavefront ϕref , separated by angleα. The residual phase variance after correction is [14,16,24]
σ2ðαÞ ¼X∞j¼Jþ1
hðajÞ2i þ 2XJj¼2
½hðajÞ2i − haðobsÞj aðrefÞj iðαÞ�; ð26Þ
where the first term of the second hand of the equality is theresidual variance due to the limited number of modes (Nollresidual), while the second term corresponds to the anisopla-natism error after J compensated modes.
The residual variance optimization in terms of the field ofview; i.e., for a given angle, is obtained when each polynomialj is verifying
hðajÞ2i − haðobsÞj aðrefÞj iðαÞ > 0: ð27Þ
This is the case when the normalized correlation function ofthe jth mode is larger than 50%. The residual variance is thenreduced. The analytic expression of this half-correlation func-tion (C1=2), introducing the polynomial radial degree n relatedto the polynomial number j by the relation 2j ¼ ðnþ 1Þðnþ 2Þ, the Bessel function of the nth order Jn, the spatialfrequency K , and the telescope diameter pupil D ¼ 2R is(see AppendixA),
C1=2ðα; n; DÞ ¼12
¼R∞
0 dhC2nðhÞ
R∞
0 dKK−14=3J2nþ1ðKÞJ0
�αhKR
�R∞
0 dhC2nðhÞ
R∞
0 dKK−14=3J2nþ1ðKÞ :
ð28Þ
The measured wavefront at angular distance α provides a sa-tisfactory fit of the on-axis wavefront when the correlation cri-terion of Eq. (28) is verified for all the correction modes. Thiscriterion allows us to select the number of useful correctionmodes for a given angular distance between the two wave-fronts. Keeping in mind that all the low orders must be firstcorrected, a higher-order mode will be corrected only if thecriterion is verified. This result is general in wavefront resi-dual variance estimation in AO using the modal analysis[25,26]. Assuming that Eq. (28) is an absolutely convergent in-tegral and that the different modes are related to the turbu-lence profile distribution only to the power spectra; i.e.,
assuming that C2nðhÞ is a constant, the numerator of
Eq. (28) is given by
Cte:Z
∞
0dKK−14=3J2
nþ1ðKÞZ
∞
0dhJ0
�αhKR
�: ð29Þ
The integral over h can be expressed by the Mellintransform
FmðsÞ ¼Z
∞
0xs−1f ðxÞdx; ð30Þ
with s ¼ 1 and using the transform properties [27,28]f ðaxÞ → a−sFmðsÞ. One obtains
Z∞
0dhJ0
�αhKR
�¼
�αKR
�−1: ð31Þ
Then, the half-correlation function is written as
Z∞
0dKK−14=3J2
nþ1ðKÞ�1 −
DαK
�¼ 0: ð32Þ
Two expressions of the Bessel functions Jnþ1 can be de-rived from the asymptotic development, at the low frequency
JνðKÞ → 1ν!
12ν
Kν; ð33Þ
and at high frequency,
JνðKÞ →ffiffiffiffiffiffiffi2πK
rcos
�K − ð2νþ 1Þ π
4
�; ð34Þ
with cos2½K − ð2νþ 1Þ π4�≃ 12 if K → ∞.
A cut-off frequency can be defined at the curve intersec-tions at low and high frequency for each radial degree,
The cut-off frequency is proportional to ðnþ 1Þ, and theZernike polynomial of order n can be evaluated by the Fouriermodes corresponding to spatial frequencies KðnÞ [29]. Theanalogy between the Zernike polynomials and Fourier modesallows the simplification of the half-correlation functions as
Z∞
0dKK−14=3δðK − KðnÞÞ
�1 −
DαcorK
�¼ 0; ð37Þ
where δ is the Dirac function, to obtain
αcor ¼D
KðnÞ≃D
nþ 1: ð38Þ
The behavior of the correlation function of the lastcorrected Zernike polynomial Nmax is given by
G. Molodij Vol. 28, No. 8 / August 2011 / J. Opt. Soc. Am. A 1735
αcor ∝DffiffiffiffiffiffiffiffiffiffiJmax
p ; ð39Þ
where the number of polynomials, Jmax ¼ ðnmaxþ1Þðnmaxþ2Þ2 ,
and Jmax > 2.An estimation of the image quality after compensation
has already been expressed by Noll from the residual phasevariance that must be lower than a given value σ2J (in rad2)[14]. After J compensated modes (J > 10), the estimation isgiven by
σ2J ¼ 0:2944 J−
ffiffi3
p2
�Dr0
�53
or σ2J ∝
�D2
J
�56
: ð40Þ
Figure 2 shows the optimal field of view αcor and the optimalimage quality related to the residual phase variance σ2J versusthe number of compensated modes J of the AO for three dif-ferent telescope apertures (1, 2, and 4m class telescope). Thedashed bottom lines indicate the image quality expected afterthe J compensated modes; the Strehl ratio is SR ¼ expð−σ2Þ.For instance, the conventional “diffraction-limited” aberrationlevel can set at the Strehl ratio value of 0.8 ; i.e., a value of σ2 ¼0:2 rad2 (indicated by the horizontal dotted line in Fig. 2).Vertical arrows defined by the intersection of the 80% Strehlratio and the σ2J functions give both the maximum order ofcompensated J of the AO and the isoplanatic patch (intersec-tion of the arrow and αcor functions). A value αiso of 5 arc secis found for the Hufnagel profile [18] that has been used toderive the different terms in the residual wavefront errorσ2ðJ; αÞ.
Two important results from this analysis are, first, the iso-planatic patch at the diffraction limit after AO compensation isindependent of the telescope aperture. Second, in the case ofa 4m class telescope, increasing the number of freedom Jfrom 450 actuators up to 1700, does not lead to a considerablyincrease of the Strehl ratio but reduce 66% of the useful fieldof view.
B. Comparison of the Finite Outer-Scale Effect on theZernike Polynomials, Angle-of-Arrivals, and Differentialof Angle-of-ArrivalsIn the context of very large aperture telescope projects,the effects of the outer scale are of great interest for AO con-sidering the values that would be about 10m. The effect of afinite spatial coherence outer scale on the covariances of theangle-of-arrival fluctuations has been discussed to determinethe influence of the model used to describe the atmosphericturbulence [11,30,31]. The derivations presented by Chassat[13] on the effect of the outer scale on the Zernike polynomialscontribute to the same study. The following analytic develop-ment allows the comparison between physical quantities ac-cessible to the measurement such as the Zernike modes, theangle-of-arrivals, and the differential of angle-of-arrivals. Thewavefront slope measurements are derived using the Primotapproach [32]. Another application is the determination of theimage quality after image stabilization [24,33]. Applying thederivative operator on a circular aperture, one obtainsthe angular covariance of the phase’s first derivative alongthe x and y directions, including the outer-scale effects (seeAppendix A)
C δφδx;yðαÞ ¼ 1:947
�Dr0
�5=3
ZL
0dhC2
nðhÞZ
∞
0dκκ−ηJ2
1ðκÞ�J0
�αhκR
�
� J2
�αhκR
���1þ
�RL0κ
�2�−11=6
; ð41Þ
where D ¼ 2R is the telescope aperture and L0 is the outerscale.
The value of η ¼ 8=3 corresponds to the angular covarianceof the angle-of-arrivals, while the value of η ¼ 2=3 goes to thederivation of the differential of angle-of-arrivals.
Figure 3 shows a comparison of the outer-scale effect be-tween the Zernike polynomials, the angle-of-arrivals, and thedifferential of angle-of-arrivals. The normalized correlationsdecrease while the outer scale L0 decreases, as indicated inFig. 3 (values of L0=R ¼ ∞, 10, and 2). Angle-of-arrivalsand the Zernike polynomials of the radial degree n ¼ 1 (tipand tilt) show the same behavior according to the value of
Fig. 2. (Color online) Optimal field of view αcor (plain lines) and optimal image quality related to the residual phase variance σ2J (dashed lines)versus the number of compensated modes J of the AO for three different telescope apertures (1, 2, and 4m class telescope indicated in blue, green,and red, respectively). Vertical axis on the left shows both the residual phase variance expressed in rad2 and the optimum field of view (arcsec) inlog units. Right vertical axis indicates the Strehl ratio related to the residual phase variance. The conventional diffraction-limit aberration level is setat a Strehl value of 0.8 (σ2 ¼ 0:2 rad2).
1736 J. Opt. Soc. Am. A / Vol. 28, No. 8 / August 2011 G. Molodij
the outer scale L0. The correlations fall 10% for the value ofL0 ¼ D at α=R ¼ 20 arcsecm−1 (Hufnagel profile model, r0 ¼10 cm at λ ¼ 0:5 μm). Figure 3 shows that the effect of the out-er scale becomes marginal, while the Zernike radial degreeincreases from the value of n ¼ 4. In the case of the seeingmonitor, the correlation fall is not significative for small val-ues of the outer scale (less than 2%). The derivation of equa-tions leads us to estimate the analytical behavior of varianceson-axis (α ¼ 0, i.e., x ¼ 0). The mean value along the two di-rections is given by (see Appendix A)
V �α ¼ 1:947
�Dro
�5=3
Z∞
0dκκ−ηJ2
1ðκÞ�1þ
�RL0κ
�2�−11=6
: ð42Þ
In the case of the Zernike coefficient variances, n is theradial degree and obtains
Vn ¼ 3:895ðnþ 1Þ�Dr0
�5=3
Z∞
0dκκ−14=3J2
nþ1ðκÞ�1þ
�RL0κ
�2�−11=6
: ð43Þ
In the case of the Zernike coefficients, μ ¼ nþ 1, η ¼ 14=3,and γ ¼ 11=6, taking into account the first principal terms ofthe serial development. For polynomials of radial degreen ¼ 1 (tip and tilt),
V1 ¼ 0:45�Dr0
�5=3
�1 − 0:77
�DL0
�1=3
þ 0:09�DL0
�2
− 0:05
�DL0
�7=3
�: ð44Þ
For polynomials of radial degree n ¼ 2 (defocus,astigmatisms),
V2 ¼ 0:023
�Dr0
�5=3
�1 − 0:39
�DL0
�2þ 0:26
�DL0
�7=3
�: ð45Þ
In the case of the mean phase derivative, μ ¼ 1, η ¼ 8=3,and γ ¼ 11=6,
V �α ¼ 1:68
�Dr0
�5=3
�1 − 0:82
�DL0
�1=3
þ 0:14
�DL0
�2
− 0:09
�DL0
�7=3
�: ð46Þ
In the case of the mean differential of the phase derivative,μ ¼ 1, η ¼ 2=3, and γ ¼ 11=6,
Vd �α ¼ 1:28�Dr0
�5=3
�1þ 0:603
�DL0
�2− 0:46
�DL0
�7=3
�: ð47Þ
The effect of the outer scale on the first derivative of thephase, the tip-tilt, and the angle-of-arrival can be expressedby a similar power law in ðD=L0Þ1=3 at the first assumption,while the effect on the second order derivative of the phase,Zernike coefficient of the second order, or differential ofthe angle-of-arrival is expressed by the same power law inðD=L0Þ2. The variances decrease with the decrease of the out-er scale, but the behavior becomes of great importance con-cerning the tip-tilt and the angle-of-arrivals for which thepartial derivative with respect to the ratio D=L0 become infi-nite at the origin. This effect is important when consideringthe AO sensor setup for which the image jitter is smaller.For instance, when L0 ≤ 5D, the angle-of-arrival and tip-tiltvalues decrease about 50%.
5. CONCLUSIONThe main result in this article is a generalization of the methodto integrands containing the product of an arbitrary number ofhypergeometric functions providing a very powerful and gen-eral technique for integral evaluation with many applicationsin physics. A general formula is given to find quantities relatedto the phase and the amplitude of the difference of two wavesthat can have different characteristics. The expression con-tains aperture filter functions that are given for determiningpiston, tilt, and higher-order aberrations on an aperture, andfor considering finite size receivers or sources. Using theseextensible aperture filter functions in the general formulas al-lows us to write the answer to many problems of practical in-terest. The Mellin transform theory can be used to evaluateevery one of these integrals which occur with any of the filterfunctions given in this paper and with any of the standardmodels of the turbulence spectra. The method is straightfor-ward, typically results in a large time savings in obtaining thesolution, and, since the answer is expressed in terms of thenatural parameters of the problem, can easily gain insight intothe solution and the underlying physics.
Appendix A: Computation of the Finite Outer-Scale Effect on the Zernike Polynomials andAngle-of-ArrivalsThe turbulent wavefront can be expressed on the set ofthe Zernike polynomials (or modes). The properties of theZernike polynomials, denoted Zj , are well described by Noll[14] whose notation is adopted. The two wavefronts comingfrom two different directions are separated by the angular dis-tance α. In Eq. (12), GðR~ρÞ is
Fig. 3. (Color online) Effect of the outer scale on the normalizedcorrelations of the Zernike polynomials n ¼ 1, 2, and 4, the angle-of-arrivals, and the differentials of angle-of-arrivals versus the angularseparation α, between the object and the reference target, normalizedby the telescope radius R. The different value of the outer scale areindicated by plain line curves for Lo ¼ ∞, diamond dot curves forLo ¼ 5D, and squaredot curves forLo ¼ D, respectively. Thederivationuse the Hufnagel profile model [18] (ro ¼ 10 cm at λ ¼ 0:5 μm).
G. Molodij Vol. 28, No. 8 / August 2011 / J. Opt. Soc. Am. A 1737
GðR~ρÞ ¼ Wð~ρÞZjð~ρÞφðR~ρÞ; ðA1Þ
where Zjð~ρÞ is defined by Eq. (20) and ~ρ is the normalizedposition vector in the aperture of radius R. Wavefronts comingfrom different directions i are expanded in the set of theZernike polynomials (piston mode removed)
φiðRi~ρÞ ¼X∞j¼2
aijZjð~ρÞ; ðA2Þ
and the Zernike expansion coefficients are
aij ¼Z
d2~ρWð~ρÞφiðRi~ρÞZjð~ρÞ; ðA3Þ
to give the angular correlations between the two wavefrontdirections and the two Zernike coefficients (j1 and j2)
ha1j1a2j2iðαÞ ¼0:033ð2πÞ−2
3k2
R2 cosΩ
ZLatm
0dhC2
nðhÞZ
d2~κQj1ð~κÞ
× Q�j2½ζðhÞ~κ�
�κR
�−113
exp½2iπαh~κ~i�: ðA4Þ
And using Eq. (21) becomes
ha1j1a2j2iðαÞ ¼ 3:895�Dr0
�5=3
R Latm0 dhC2
nðhÞζ−1ðhÞIð αhR1ðhÞ ; ζðhÞÞR Latm
0 dhC2nðhÞðR1ðhÞ
R1ð0ÞÞ5=3;
ðA5Þ
where r0 is the Fried parameter, D is the telescope diameter,C2
nðhÞ is the turbulence profile, h is the altitude along the pro-pagation path from the star to the telescope, and taking intoaccount the extensible aperture function (artificial star wave-front propagation case),
1;2 are coefficients that depend on the polynomialparity and the azimuthal frequency given in Table 1. A power-ful method for evaluating integrals has been described byShelton and Sasiela [19,34] that applies to integrals whose in-tegrands are the product of two generalized hypergeometricfunctions. These integrals, which can be transformed into aMellin–Barnes integral in the complex plane, can be ex-pressed as a finite sum of generalized hypergeometric func-tions. Extended derivations for the study of the specificapplication to anisoplanatism for the solar extended field ofview have already been presented [24], as well as an analysisof the laser guide star anisoplanatism limitations [16]. Thestudy of the effects of a finite outer scale led us to solvethe following integral in the respective cases of the Zernikepolynomials, the angle-of-arrivals, and the differential angle-of-arrivals. The integrals to solve are
Iðx; y; ηÞ ¼Z
∞
0dκκ−η½1þ ðκyÞ−2�−γJ2
μðκÞJνðκxÞ; ðA10Þ
with x ¼ αhR , y ¼ Lo
R , and γ ¼ 116 .
To determine the Mellin–Barnes integral, the parametersare, respectively, in the case of the Zernike decompositionof radial degree n and azimuthal frequency m, μ ¼ nþ 1, ν ¼0 or 2m, and η ¼ 14=3; in the case of the correlation of theangle-of-arrival, μ ¼ 1, ν ¼ 0 or 1, and η ¼ 8=3; in the caseof the correlation of the differential of angle-of-arrivals,μ ¼ 1, ν ¼ 0 or 1, and η ¼ 2=3,
Iðx;y;ηÞ¼ 1
ð2iπÞ2
×Z þi∞
−i∞
Z þi∞
−i∞Γ�−t;γþ t;−t−sþ 1
2ð−ηþ1Þþμ;tþsþ η2 ;sþ ν
2
γ;μþ tþsþ 12ðηþ1Þ;tþsþ 1
2ðηþ1Þ;1−sþ ν2
�
×1
2ffiffiffiπp�x2
�−2s
y−2tdsdt; ðA11Þ
using the notation of the gamma function of variable x, ΓðxÞ,
Table 1. Parity Rules
Kþ1;2 m1 ¼ 0
m1 ≠ 0 andj1 even
m1 ≠ 0 andj1 odd
m2 ¼ 0 1ffiffiffi2
p0
m2 ≠ 0 and j2 evenffiffiffi2
p1 0
m2 ≠ 0 and j2 odd 0 0 −1
K−
1;2 m1 ¼ 0m1 ≠ 0 andj1 even
m1 ≠ 0 andj1 odd
m2 ¼ 0 0 0 0m2 ≠ 0 and j2 even 0 1 0m2 ≠ 0 and j2 odd 0 0 1
1738 J. Opt. Soc. Am. A / Vol. 28, No. 8 / August 2011 G. Molodij
Γ�x1; x2;…:; xny1; y2;…:; ym
�¼ Γðx1ÞΓðx2Þ…:ΓðxnÞ
Γðy1ÞΓðy2Þ…:ΓðymÞ:
In the integral numerator, gamma functions show the distinctsets of singular poles in the complex plane. The functions canbe expressed [34] as
ΓðsÞ ¼Z
∞
Odκ expð−κÞκs−1
¼X∞n¼0
ð−1Þnn!
1sþ n
þZ
∞
1dκ expð−κÞκs−1; ðA12Þ
with ReðsÞ > 0.Pole-residue integration is used to evaluate the Mellin–
Barnes integral. To apply this method, the integration pathmust be closed in the complex plane where the integral con-verge absolutely. Singularities come only from the gammaratio of the numerator in Eq. (A11). Following the Cauchy the-orem solving procedure, one selects the contour of integrationon the complex plane. The combinations of distinct bi-poles are
1. s ¼ −qþ pþ 12 ð−ηþ 1Þ þ μ and t ¼ q
2. s ¼ −p − q − η2 and t ¼ q
3. s ¼ −p −ν2 and t ¼ q
4. s ¼ pþ qþ γ þ 12 ð−ηþ 1Þ and t ¼ −q − γ
5. s ¼ −pþ qþ γ − η2 and t ¼ −q − r
6. s ¼ −p −ν2 and t ¼ −q − r
7. μþ 1 ¼ −q − p (impossible case with integers)8. s ¼ −p −
ν2 and t ¼ pþ qþ μþ 1
2 ðν − ηþ 1Þ9. s ¼ −p −
ν2 and t ¼ p − qþ 1
2 ðν − ηÞ.
The contribution of the bipoles depends on the relative sizebetween x and y, denoting the Mellin–Barnes integral con-verge only if y−2tðx2Þ−2s converge. The direction of the path clo-sure depends on the integrand behavior at infinity (whenjtj → ∞). To determine the direction path closure, Sasiela de-fines a criterion denoted Δ [34]. Let h be an integral over κ,
hðκÞ ¼Z
dκκ−sΠA
i¼1Γ½ai þ αis�ΠBj¼1Γ½bj − βjs�
ΠCk¼1Γ½ck þ γks�ΠD
m¼1Γ½dm − δms�; ðA13Þ
Δ ¼XAi¼1
αi þXDm¼1
δm −
XBj¼1
βj −XCk¼1
γk: ðA14Þ
If Δ ¼ 0, the integral converges regardless of the directionof the path closure.
If Δ > 0, the direction of the path closure must be on theleft, i.e., ReðxÞ ≤ 0. If Δ < 0, the direction of the path closuremust be on the right, i.e., ReðxÞ ≥ 0.
Using the convergence criteria of Eq. (A14), one determinesthat Δs ¼ 0 and Δt ¼ −2. The path closure, regardless of thedirection in the complex plane s, must be right in the t com-plex plane. Considering the situation Lo ≥ R (i.e., y > 1), onlybipoles 1, 2, 3, and 8 participate in the derivation. Case 9 cor-responds to an outer scale smaller than the telescopeaperture.
The serial development to each of the possible bipoles is
I1ðx; yÞ ¼1
2ffiffiffiπpX∞p¼0
X∞q¼0
ð−1Þpþq
p!q!
�x2
�2q−2p−2μþη−1
y−2qΓ"γ þ q; pþ μþ 1
2 ;−qþ pþ μþ 12 ðν − ηþ 1Þ
γ; qþ 2μþ 1; qþ μþ 1; pþ νþ 1
#; ðA15Þ
I2ðx; yÞ ¼1
2ffiffiffiπpX∞p¼0
X∞q¼0
ð−1Þpþq
p!q!
�x2
�2qþ2pþη
y−2qΓ"
γ þ q; pþ μþ 12 ;−q − pþ 1
2 ðν − ηÞγ; qþ pþ 1þ 1
2 ðνþ ηÞ; μ − pþ 12 ;−pþ 1
2
#; ðA16Þ
I3ðx; yÞ ¼1
2ffiffiffiπpX∞p¼0
X∞q¼0
ð−1Þpþq
p!q!
�x2
�2pþν
y−2qΓ"
γ þ q; pþ μ − qþ 12 ðν − ηþ 1Þ; q − pþ 1
2 ðν − ηÞγ; μþ q − pþ 1
2 ðν − ηþ 1Þ; q − pþ 12 ðν − ηþ 1Þ; pþ νþ 1
#ðA17Þ
I8ðx; yÞ ¼1
2ffiffiffiπpX∞p¼0
X∞q¼0
ð−1Þpþq
p!q!Γ"−μ − q − pþ 1
2 ðν − η − 1Þ; pþ μþ qþ γ þ 12 ðν − ηþ 1Þ; qþ 1
2 þ μγ; 2μþ qþ 1; μþ qþ 1; pþ νþ 1
#�x2
�2pþν
y−2q−2p−νþη−1−2μ: ðA18Þ
G. Molodij Vol. 28, No. 8 / August 2011 / J. Opt. Soc. Am. A 1739
One obtains a combination of the previous serial develop-ments depending of the values of x ¼ αh
R and y ¼ LoR .
If x ≤ 2 then x < 2y is always verified with y > 1, andthen Iðx; yÞ ¼ I2ðx; yÞ þ I3ðx; yÞ þ I8ðx; yÞ.
If x > 2 and x < 2y, then Iðx; yÞ ¼ I1ðx; yÞ þ I8ðx; yÞ.Otherwise, there are no solutions.In the case of variances (estimated on axis with α ¼ 0, i.e.,
x ¼ 0), the angle-of-arrivals and Zernike polynomials show theidentical Mellin transform expression
IðxÞ ¼ 12πi
Zcþi∞
c−i∞GmðtÞHmð−η − tþ 1Þx−tþη−1dt; ðA19Þ
where hðxκÞ ¼ ½1þ ðxκÞ−2�γ with x ¼ L0=R, and gðκÞ ¼ J2μðκÞ.
To derive the serial development
IðxÞ ¼ 12
ffiffiffiπpX∞p¼0
ð−1Þpp!
��L0
R
�−2p
× Γ�−pþ μþ 1
2 ð1 − ηÞ; pþ η2 ; pþ γ
γ; μþ pþ η2 þ 1
2 ; pþ η2 þ 1
2
�þ�L0
R
�−2p−2μþη−1
× Γ�pþ μþ 1
2 ;−p − μþ 12 ðη − 1Þ; pþ μþ γ þ 1
2 ð1 − ηÞγ; 2μþ pþ 1; μþ pþ 1
��;
ðA20Þ
with Lo ≫ R.
ACKNOWLEDGMENTSSpecial thanks to G. Rousset who initiated this work severalyears ago. I wish to thank particularly the referees for the ac-curate reading of the paper and for providing me very con-structive comments.
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