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E. Tatulli and A. N. Ramaprakash

Laser tomography adaptive optics: a performance study Eric Tatulli* and A. N. Ramaprakash Inter-University Centre for Astronomy and Astrophysics, Ganeshkhind, Pune 411 007, India *Corresponding author: [email protected] Received July 10, 2013; revised October 7, 2013; accepted October 10, 2013; posted October 10, 2013 (Doc. ID 193349); published November 8, 2013 We present an analytical derivation of the on-axis performance of adaptive optics systems using a given number of guide stars of arbitrary altitude, distributed at arbitrary angular positions in the sky. The expressions of the residual error are given for cases of both continuous and discrete turbulent atmospheric profiles. Assuming Shack–Hartmann wavefront sensing with circular apertures, we demonstrate that the error is formally described by integrals of products of three Bessel functions. We compare the performance of adaptive optics correction when using natural, sodium, or Rayleigh laser guide stars. For small diameter class telescopes (≲5 m), we show that a small number of Rayleigh beacons can provide similar performance to that of a single sodium laser, for a lower overall cost of the instrument. For bigger apertures, using Rayleigh stars may not be such a suitable alternative because of the too severe cone effect that drastically degrades the quality of the correction. © 2013 Optical Society of America OCIS codes: (010.7350) Wave-front sensing; (010.1290) Atmospheric optics; (070.0070) Fourier optics and signal processing; (000.3860) Mathematical methods in physics; (140.0140) Lasers and laser optics. http://dx.doi.org/10.1364/JOSAA.30.002482

1. INTRODUCTION The concept of using artificial laser guide stars (LGSs) for adaptive optics (AO) systems [1,2] has been proposed to increase sky coverage by enabling partial correction of the effects of the atmospheric turbulence in regions where no bright ‘natural guide stars are present in the vicinity of the astrophysical source of interest. In such an instrumental configuration the fundamental limits preventing a perfect correction of the incoming corrugated wavefront have three origins, the latter being specific to the use of LGS: 1. The inability of the wavefront sensor (WFS) to probe and/or the deformable mirror (DM) to correct some (often the high) frequency components of the turbulent wavefront (the so-called fitting error), together with the presence of photon and detector noises associated with the WFS measurements. 2. The spatial and temporal decorrelation between the science and guide star wavefronts, when the guide star is located off-axis and when the correction is applied with a temporal delay due to the finite temporal frequency of the AO control loop. 3. The spherical nature of the LGS wavefront because of the finite altitude of the artificial spot that drives its coneshaped beam to cross only a fraction of the turbulence seen by the science target, resulting in an additional term in the error budget known as focus anisoplanatism [3] or most commonly described as the cone effect [4]. In order to cancel the latter effect that severely reduces the performance of AO systems, it has been proposed to simultaneously use several LGSs located at different angular positions in the sky and to perform a 3D mapping of the turbulent volume [4]. For this so-called laser tomography adaptive optics (LTAO) technique [5], each LGS is associated to a dedicated WFS, and the corrugated wavefront estimated 1084-7529/13/122482-20$15.00/0

from the 3D-mapped turbulence is compensated with a single DM conjugated to the telescope pupil, thus providing a potentially important correction of the atmospheric effects but over a narrow field of view. Generating artificial spots in the sky can be achieved either by Rayleigh backscattering for low altitude atmospheric layers (≲20 km) or by excitation of sodium atoms in the mesospheric sodium layer located at ≃90 km. Although the first solution requires only mainstream—hence economical—laser technology over a large range of wavelengths [6], making use of such Rayleigh stars has been mostly abandoned as their low altitude prevents good correction of the cone effect, especially for large apertures [7]. Considering its low cost, the potential of Rayleigh LGS, however, deserves to be quantified in perspective of the financial benefits. On the contrary, sodium stars are often preferred because of the less severe cone effect. However, they necessitate custom-made state-of-the-art expensive lasers [8] that drastically increase the budget of the AO system, all the more since several LGSs are contemplated. The capabilities of the LTAO technique have been investigated through bench demonstrators [9] and by means of performance simulations for specific AO systems with large aperture (GALACSI-VLT [10], GMT [11], ATLAS-ELT [12]), but no generic theoretical study has been published so far. The aim of our paper is thus twofold: in the first part, we provide in Sections 2 and 3 a formal derivation of the performance of LTAO, taking into account in a unified modeling the effects of focus anisoplanatism, incomplete wavefront sensing, as well as spatial and temporal decorrelation between the science and guide star wavefronts, for both continuous and discrete profiles of turbulence. In the second part, we use this analytical framework to quantitatively study in Sections 4 and 5 the cases of AO systems using one or several LGSs. We finally presents in Section 6 a comparison of the © 2013 Optical Society of America

E. Tatulli and A. N. Ramaprakash

Vol. 30, No. 12 / December 2013 / J. Opt. Soc. Am. A

performance that can be expected when using sodium or Rayleigh lasers with different existing AO systems on telescopes with apertures ranging from 3 to 10 m.

2. BACKGROUND FORMALISM AND UNDERLYING ASSUMPTIONS A. Wave Propagation and Bessel Functions Integrals involving the product of Bessel functions have been shown to be an important feature of electromagnetic field propagation through atmosphere [13,14]. Following the notation of Hu et al. [13], we introduce the definition of functions H2J and H3J that will be convenient to express the results of our analytical derivations: Z H2J
s; n1 ; n2 ; a; b 

0

Z H3J
s; n1 ; n2 ; n3 ; a; b; c 

∞

∞

0

x−s J 2n1
axJ n2
bxdx;

(2) where J n1 ;n2 ;n3 are Bessel functions of the first kind of order n1 , n2 , and n3 , respectively, and s; a; b; c are parameters of H2J and H3J functions. Integrals of that form are related to the Mellin transform [15], and formal evaluations involving gamma and hypergeometric functions [16] can be performed in some specific cases, as provided by Gradshteyn et al. [17] (see Eq. 6.578 #1) and by Tyler [18]. B. Independent Tip/Tilt Correction Wavefront sensing with monochromatic LGSs is unable to measure the random shift (tip/tilt) of the image because of the inverse return of light principle. Several concepts have been proposed to solve this indeterminacy, such as making simultaneous use of two small auxiliary telescopes [19] or two LGSs [20], by taking advantage of the properties of polychromatic LGSs [21] or by adding to the whole AO system a specific instrument dedicated to estimation of the image displacement by pointing a nearby natural guide star [22]. Since our analysis focuses on the performance of LGS AO, that is, the correction of higher-order modes than tip and tilt, we assume in the following that these are estimated independently and fully corrected. In order to take into account partial tip/tilt correction, a quadratic error must be added to the error budget following, e.g., the formalism of Sandler [23] that models the atmospheric tip/tilt error (influence of higher modes on the estimation of tip/tilt [24]), tip/tilt anisoplanatism error, and photon/detector noise associated with the tip/tilt measurements. C. Science Star Turbulent Wavefront We define Φ
r as the turbulent phase of the plane wavefront arising from the science star. Using Zernike polynomials, the piston/tip-tilt removed science phase can be written as Φ
Rρ 

∞ X

ϕj Z j
ρ

The piston mode is also not considered as it is irrelevant for AO correction and wavefront sensing issues. The statistics of the turbulent science phase is characterized by the covariance matrix Cov
ϕ  hϕϕT i, where hi denotes the statistical average and T is the transpose operator. Following Noll description [25], the turbulence variance σ 2ϕ , that is, the trace of the covariance, is
5 D 3 ; TrfCov
ϕg  σ 2ϕ  0.135 r0

(3)

j3

with R the radius of the telescope aperture, and ρ  r∕R, the polynomials being defined over the unit radius circle.

(4)

where D  2R is the diameter of the telescope and r 0 is the Fried parameter defined at the zenith as [26] 2

(1)

x−s J n1
axJ n2
bxJ n3
cxdx;

2483

r0  4

2

0.033
2π−3

 2 2π λ

0.023

Z

∞ 0

3 −3 5

C 2n
hdh5

:

(5)

C 2n
h is the atmospheric structure constant of the refractive index along the altitude h above the telescope. D. Spherical LGS Wavefronts: The Cone Effect We call Φlgs
r; αp  the turbulent phase of the spherical wavefronts coming from the N lgs LGS located at respective angular position αp , p ∈ 1::N lgs  that are used to probe the atmospheric turbulence. The portion of atmosphere crossed by the laser beams—therefore the turbulent LGS phase—depends on αp . The spatial covariance Blgs Φ
Rρ of the LGS turbulent phase that characterizes its statistical properties is, however, independent of this angular location and can be written as Blgs Φ
Rρ  hΦlgs
Rρ1  ρ; αp Φlgs
Rρ1 ; αp i
2 Z  ZL L 2π  n
Rζ
hρ1  ρ; αp dh n
Rζ
hρ1 ; αp dh λ 0 0 (6)


2π λ

2 Z 0

L

BhΔn
Rζ
hρdh;

(7)

where n is the refractive index and BhΔn is the covariance of its fluctuation for the turbulent layer located at the altitude h and of infinitesimal thickness δh, and assuming that these layers are statistically independent (small perturbations and nearfield approximations [27]). Due to the spherical nature of the LGS wavefront (cone effect), the fraction of turbulence ζ
h seen by the LGS beam at altitude h is ζ
h 
L − h∕L with L the altitude of the LGS, as shown in Fig. 1 (left). The power spectrum W lgs Φ
κ of the LGS phase is by definition the Fourier transform of its spatial covariance and thanks to Eq. (7) can be written as W lgs Φ
κ 



2 Z L 2π 1 κ h dh: W Δn 2 λ Rζ
h 0 Rζ
h

(8)

Under Kolmogorov statistics hypothesis [28,29], the refractive index fluctuation power spectrum W hΔn
κ is given by

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z science star incoming wavefront LGS

#k

ρk

L

θk

Turbulent medium

2R ζ (h)

αp r h

R Telescope

Normalized radius

Fig. 1. Left: sketch of LGS AO observations. The angular location of the laser spot is αp , and its height is L. For a given altitude h, the laser beacon light crosses a turbulence portion of radius ζ
hR, with ζ
h 
L − h∕h. Right: representation of the Shack–Hartmann (SH) subapertures, in polar coordinates. The kth subaperture is located at a normalized radius ρk and an angle θk .

2

BΦ
Rρ; Rρ1 ; αp ; τ Φlgs

11

W hΔn
κ  0.033
2π−3 jκj− 3 C 2n
h
2 λ C 2
h 11 −5  0.023r 0 3 jκj− 3 R ∞ n2 ; 2π 0 C n
hdh

(9)

 hΦ
Rρ1  ρ; tΦlgs
Rρ1 ; αp ; t  τi
2 Z  ZL L 2π  n
Rρ1  ρdh n
Rζ
hρ1  hαp  τv
hdh λ 0 0

and W lgs Φ
κ takes the final form

(12)

R
5 5 R 3 −11 0L ζ
h3 C 2n
hdh 3 R W lgs
κ  0.023 jκj : Φ ∞ 2 r0 0 C n
hdh

(10)

The ratio of the integrals over the altitude captures the cone effect due to the finite altitude of the LGS. In the case of a plane wavefront, we have L  ∞ and ζ
h  1; hence the ratio is equal to 1 and we obtain the definition of the classical Kolmogorov phase power spectrum. Finally we describe the LGS phase over the Zernike polynomial basis as follows:

Φlgs
Rρ; αp  

∞ X

ϕlgs j
αp Z j
ρ:

(11)

j1

E. Control Loop Delay An AO loop works at a finite speed (roughly a few hundred hertz), which translates into a time delay τ between the observation of the scientific source and the actual correction of the atmospheric perturbations from the guide star. In such a case, the science star turbulent phase Φ
r; t taken at given time t will be corrected from the LGS phase Φlgs
r; αp ; t  τ taken at a time t  τ. Under Taylor hypothesis of “frozen turbulence,” this time delay can be transformed into a spatial shift Δρ  τv
h, where v
h is the wind speed vector for the altitude h. The crossed covariance between the science star and guide star phases can thus be computed as:




2 Z L 2π BhΔn
Rρ1 1 − ζ
h  Rρ − hαp − τv
hdh: λ 0

(13)

We emphasize that, at the difference of the plane and spherical wavefront phase covariances, the cross covariance is a nonstationary process since it depends on the location ρ1 where this quantity is computed from. For describing the wind associated with the turbulent layers, Bufton [30] has provided an empirical law for the wind speed modulus:  
h − 9.42 v
h  5  30 exp − ; 4.82

(14)

where the numbers outside the brackets are in meters per second. From the wind speed average v¯ , one can estimate the coherence time of the turbulence t0 , using the definition of Greenwood [31]: t0  0.314

r0 : v¯

(15)

F. Wavefront Sensing We assume that identical Shack–Hartmann (SH) WFSs [32] are associated to every LGS beam. We call M s the number of subapertures of each SH that will therefore provide 2M s slope measurements corresponding to the LGS turbulent phase. We denote s
αp   sx
αp ; sy
αp  as such slope measurements, in the x and y directions. Considering the

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Vol. 30, No. 12 / December 2013 / J. Opt. Soc. Am. A

kth subaperture, the SH provides the derivative of the LGS phase as follows [32]: Z

∂Φlgs
r; αp  2 dr ∂x; y subapk Z λR ∂  Φlgs
Rρ; αp d2 ρ: 2πAs subapk ∂x; y

sx;y k
αp  

λ 2πAs

(16)

(17)

Note that D∞ is block diagonal, the number of blocks being equal to the number of LGS/AO used. Each block is made of two submatrices Dx ; Dy  that account for the slope measurements in both directions; that is, Dx;y kj

λR  2πAs

Z

∂Z j
ρ 2 λR d ρ 2πA ∂x; y subapk s




Z Πks

 ∂Z j
ρ 2 R d ρ; ρ Rs ∂x; y (18)

where Πks

R∕Rs ρ is a function of the kth subaperture and Rs is its characteristic size. It can be rewritten in the form Πks

R∕Rs ρ  Πs

R∕Rs ρ − ρk , where ρk  ρk ; θk  is the normalized coordinate vector of the kth subaperture, with respect to the center of the telescope aperture, as shown in Fig. 1 (right). For a circular subaperture of radius Rs , we have ˆ ks
κ of the kth subaperAs  πR2s and the Fourier transform Π ture can be written as


 R ρ − ρk  exp−2iπρ·κ d2 ρ Rs     J 2π Rs jκj R Rs 1  exp−2iπρk ·κ : jκj R

ˆ ks
κ  Π

Z

Πs

(19)

In such a case, the elements of the interaction matrix can be computed formally in terms of integrals of products of three Bessel functions, as demonstrated in Appendix A.3. Using notations of Section 2.A, we have Dxkj 

λ s βx
θ H3J
0; 1; n  1; jmj − 1; Rs ∕R; 1; ρk  2πRs n;m jmj−1 k − βxjmj1
θk H3J
0; 1; n  1; jmj  1; Rs ∕R; 1; ρk ;

Dykj 

(20)

λ s βy
θ H3J
0; 1; n  1; jmj − 1; Rs ∕R; 1; ρk  2πRs n;m jmj−1 k  βyjmj1
θk H3J
0; 1; n  1; jmj  1; Rs ∕R; 1; ρk ;

(21)

where n and m are, respectively, the radial degree and the azimuthal frequency associated to the jth Zernike polynomial and sn;m , βx;y jmj1 are defined by sn;m 

3n ijmj
−1 2

p p 2 if m ≠ 0 n1 ; 1 if m  0

cos
jmj  1θk ; − sin
jmj  1θk  if m ≥ 0 : sin
jmj  1θk ; cos
jmj  1θk  if m < 0 (23)

As is the area of the subaperture, and λ is the wavelength of the AO WFS path. From Eqs. (16) and (11), we can introduce the interaction matrix D∞ that converts the LGS phase Zernike coefficients into SH slope measurements: s  D∞ ϕlgs :

βx;y jmj1;k
θk  

2485

(22)

When using LGS beacons, the SH will not be sensitive to the tip/tilt modes of the LGS phase. As a result the tip and tilt contributions to the slopes must be removed, such that the effective measured slopes sˆ are given by  λ ϕlgs ; πR 1


sˆ x  sx −

 λ ϕlgs ; πR 2


sˆ y  sy −

(24)

lgs where ϕlgs 1 and ϕ2 are, respectively, the tip and tilt Zernike coefficients of the LGS turbulent wavefronts.

G. Perfect Deformable Mirrors For the sake of simplicity we assume in the following that the DM is able to perfectly reproduce the shape of the wavefront provided by the WFSs. In practice there is, however, a mismatch between the desired wavefront and the surface that the mirror will eventually take, since the number of actuators that shape the surface of the mirror is not infinite. This mismatch can be modeled by taking into account the projection of the slopes onto the DM modes, that is, the actuators’ responses. We refer to the work of Wallner [33] (single guide star case) and Tokovinin et al. [34] (multiple guide stars case) for a modeling of the problem that includes this effect. H. Wavefront Reconstruction and Residual Phase Error ~ We call Φ
r the estimated turbulent phase from the ~ its related Zernike coefficients slope measurements and ϕ vector. The residual phase variance is by definition the variance of the phase difference integrated over the pupil of the telescope: σ 2res 

Z

2 2 ~ Πp
ρhjΦ
Rρ − Φ
Rρj id ρ;

(25)

where Πp
ρ is the unitary pupil function. ~ from the measurements s is a linear The computation of Φ fitting process. We introduce M the so-called control matrix [33] representing this process. We thus can write the following relationship: ~  M
ˆs  ϵ; ϕ

(26)

where ϵ is the additive (i.e., photon, detector) noise associated to the slopes. Data cosmetics (flat-field, dark current, etc.) are not considered in this paper since these effects are assumed to be removed through proper calibration. If we assume an aperture without central obstruction, standard Zernike polynomials form an orthonormal basis and Eq. (25) simplifies as ~ 2 iatm;ϵ  h‖ϕ − M
ˆs  ϵ‖2 iatm;ϵ ; σ 2res  h‖ϕ − ϕ‖

(27)

where hiatm;ϵ is the average over both the atmosphere and the additive noise statistics. The explicit form of M will be investigated in Section 3.C.

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E. Tatulli and A. N. Ramaprakash

3. COMPUTATION OF THE RESIDUAL PHASE ERROR The aim of this section is threefold: first we provide the formal expression of the residual phase error in the general case of multiple LGS AO corrections and continuous turbulent atmospheric profile. However, performing tomography of the turbulence requires us to describe the atmosphere as thin discrete turbulent layers located at specific heights. In this respect, we also provide the computation of the residual error using an independent matrix-oriented approach. From this latter modeling, we finally derive the expression of the optimal control matrix M that enables us to minimize the residual error. A. General Analytical Approach With further hypothesis that atmospheric and additive noises are independent, the matrix expression of the previous equation is σ 2res  Trfh
ϕ−M
ˆs ϵ
ϕ−M
ˆs ϵT iatm;ϵ g  TrfhϕϕT iMhˆssˆ T iM T −hϕˆsT iM T −MhˆsϕT iMhϵϵT iM T g  TrfCov
ϕMCov
ˆsM T −2Cov
ϕ; sˆ M T MCov
ϵM T g: (28)

Cov
s denotes the covariance of the slope measurements. As sˆ is the concatenation of x and y slopes for each LGS located at αp , the elements of the matrix result in the computation of yy xy xx three moments C xx sxk
αp ˆsxl
αq i, s , C s , and C s with C s  hˆ yy y y x x C s  hˆsk
αp ˆsl
αq i, and C xy  hˆ s
α ˆ s
α which, s q i, k p l according to Eq. (24), leads to
x x C xx s  hsk
αp sl
αq i 


−

λ πR





λ πR

2





λ πR

2

(29)

lgs hϕlgs 2
αp ϕ2
αq i

 λ lgs y hsyk
αp ϕlgs 2
αq i  hsl
αq ϕ2
αp i; πR

y x C xy s  hsk
αp sl
αq i 

−

lgs hϕlgs 1
αp ϕ1
αq i

 λ lgs x hsxk
αp ϕlgs 1
αq i  hsl
αq ϕ1
αp i; πR

y y C yy s  hsk
αp sl
αq i 

−

2

(30)

lgs hϕlgs 1
αp ϕ2
αq i

 λ lgs y hsxk
αp ϕlgs 2
αq i  hsl
αq ϕ1
αp i: πR

(31)

The formal expressions of the moments involved in the computation of Cov
s are given in Appendix B. For the case of SH circular subapertures, the moments can be written using H2J and H3J functions:

8 x 9 hs
α sx
α i > > > < k p l q > = hsyk
αp syl
αq i > > > : hsx
α sy
α i > ; k

p

l

q


5
2 0.0493 D 3 λ  R∞ 2 r Rs C
hdh 0 n 0 28 9 > >1> > ZL 6< = 5 pq 2 × dhζ
h3 · C n
h6 1 4> >HJ2
8∕3; 1; 0; Rs ∕R; ρkl
h 0 > > : ; 0 8 9 3 pq cos
2θkl
h > > > > < = 7 pq 7 (32) − − cos
2θpq kl
h HJ2
8∕3; 1; 2; Rs ∕R; ρkl
h5; > > > > : ; pq sin
2θkl
h

8 lgs 9 hϕ
α ϕlgs
α i > > > < 1 p 1 q > =
αp ϕlgs
αq i hϕlgs 2 2 > > > > : lgs ; hϕ1
αp ϕlgs 2
αq i
5 7.791 D 3  R∞ 2 0 C n
hdh r 0

28 9 > >1> > L 6< = 5 2 6 3 × dhζ
h · C n
h4 1 HJ2
14∕3; 2; 0; 1; ρpq
h > > 0 > : > ; 0 8 9 3 cos
2θpq  > > > > < = 7 (33) − − cos
2θpq  HJ2
14∕3; 2; 2; 1; ρpq
h7 5; > > > > : ; sin
2θpq  Z

8 x 9 hsk
αp ϕlgs > 1
αq i > > > > > > > > < hsy
α ϕlgs
α i > = p q k 2 > > > hsxk
αp ϕlgs > > 2
αq i > > > > > : y ;
α i hsk
αp ϕlgs q 1
5
 0.620 D 3 λ  R∞ 2 r Rs C
hdh 0 n 0 28 9 > >1> > > > > 6> ZL = 6> 5 6 2 dhζ
h3 · C n
h6 × HJ3
11∕3; 1; 2; 0; Rs ∕R; 1; ρpq k
h > 6> 0 0 > > > > > 4> > : > ; 0 8 9 3 > cos
2θpq k  > > > > > > > 7 > < − cos
2θpq  > = 7 k 7 pq (34) − ∕R; 1; ρ
h HJ3
11∕3; 1; 2; 2; R 7; s k pq > > 7  sin
2θ > > k > > 5 > > > > : ; sin
2θpq k  where ρkl ; θkl   ρl − ρk , ρpq
h; θpq  
h∕Rζ
h
αq − αl , pq pq pq ρpq k
h; θk
h 
h∕Rζ
h
αq − αp  − ρk , and ρkl
h;θkl
h 
h∕Rζ
h
αq −αp ρl −ρk . Similarly Cov
ϕ; s represents the cross correlation between the slopes and the tip/tilt removed science star turbulent phase. The elements C xsϕ and C ysϕ of the matrix are defined by

E. Tatulli and A. N. Ramaprakash

Vol. 30, No. 12 / December 2013 / J. Opt. Soc. Am. A


C xsϕ  hsxk
αp ϕj i −
C ysϕ  hsyk
αp ϕj i −

 λ hϕlgs 1
αp ϕj i; πR

(35)

 λ hϕlgs 2
αp ϕj i: πR

(36)

The computation of these moments is provided in Appendix B. In the specific case of SH circular subapertures, their expression involves the H3J function: (

hsxk
αp ϕj i

)

hsyk
αp ϕj i


5
 Z L 0.310 D 3 λ dhC 2n
h 2 Rs 0 0 C n
hdh r 0 ) "( x βjmj−1
θpk
h HJ3
11∕3; 1; × βyjmj−1
θpk
h

 sn;m R ∞

(

hϕlgs 1
αp ϕj i

SH subapertures due to the parallax effect and the nonzero thickness of the layer where the spot is created. This additional effect that varies with the radial location of the LGS can be taken into consideration by replacing previous equations with those of, e.g., Béchet et al. [35] [see Eq. (6) of their paper]. B. Discrete Turbulent Layers: Matrix Approach We now assume that the turbulent medium can be modeled by a discrete sum of N el equivalent, statistically independent turbulent layers of thickness Δh, as sketched in Fig. 2. In such a case, Eqs. (3) and (11) can be, respectively, rewritten as Φ
Rρ 

N el ∞ X X

ϕj
hk Z j
ρ;

(41)

ϕlgs j
αp ; hk Z j
ρ;

(42)

j4 k1

n  1; jmj − 1; ζ
hRs ∕R; 1; ρpk
h ) ( x −βjmj1
θpk
h HJ3
11∕3; 1; n  1;  βyjmj1
θpk
h # jmj  1; ζ
hRs ∕R; 1; ρpk
h ;

2487

Φlgs
Rρ; αp  

N el ∞ X X j1 k1

(37)

)

hϕlgs 2
αp ϕj i


5 Z 3.986 D 3 L dhζ
h−1 C 2n
h 2 0 0 C n
hdh r 0 ) "( x βjmj−1
θp  HJ3
14∕3; 2; n  1; jmj − 1; ζ
h; 1; ρp
h × βyjmj−1
θp  ) # ( x −βjmj1
θp  p HJ3
14∕3; 2; n  1; jmj  1; ζ
h; 1; ρ
h ;  βyjmj1
θp 

 sn;m R ∞

(38)

where ϕ
hk  and ϕlgs
αp ; hk  are the Zernike coefficients for, respectively, the science and LGS phases of the kth turbulent layer. For each layer, the outer part of all the LGS cone beams together defines the limits of a so-called metapupil [36], which covers the turbulence crossed by both the science and LGS wavefronts at that layer. If αmax is the largest angular location of the LGS network, the size of the metapupil RM
h is defined as

RM
h

z

Equivalent turbulent layers

(39)


2 2
2 2 π σ e− 4X T σ 2d  p ; 2 XD N 3 ph

(40)

where N ph is the number of photons per subaperture, σ e− is the detector noise rms per pixel, and X T and X D are the full width half-maximum (in pixels) of, respectively, the turbulent and diffraction-limited subaperture image spots. As it is beyond the scope of this paper, the previous equations do not take into account the effect of the laser spot elongation on the

 R  : R  h jαmax j − RL

if jαmax j ≤ RL if jαmax j > RL

:

(43)

We call ΦM and ϕM
hk  the phase and its associated Zernike coefficients defined over the metapupils of each turbulent layer. Ragazzoni et al. [36] have shown that there exist linear procedures (i.e., matrices) that allow one to deduce the

where ρpk
h; θpk
h 
hαq ∕R  ζ
hρk and ρp
h; θp  
hαq ∕R. Finally, Cov
ϵ represents the additive noise covariance. Assuming identical noises for all SH and that the noises are independent between two different subapertures, the covariance matrix can be rewritten Cov
ϵ  σ 2ϵ · Id, where Id is the identity matrix and σ 2ϵ is the quadratic sum of the photon (σ 2p ) and detector noises (σ 2d ). Rousset [32] has given an expression for both noises, in the case of SH WFSs:

2 π 1 XT 2 σ 2p  p ; 2 N ph X D



8
4000 when the MMSE LGS method is used in place of α ≃ 1500 for the SVD/MMSE NGS reconstruction techniques. Using the MMSE reconstruction in order to optimize LGS AO correction, however, requires a priori knowledge of both the altitude of the turbulent layers and the location of the guide star, parameters that are difficult to estimate precisely and can also slowly vary with time. Figure 4 (bottom) investigates (for the one-layer atmospheric model) the robustness of the technique to an error in the estimation of the layer

E. Tatulli and A. N. Ramaprakash

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Fig. 4. Top: residual error as a function of the number of subapertures (left) and the angular separation between the science and guide stars (right). Bottom: residual error as a function of the estimated turbulence layer altitude (left) and LGS angular position (right). The correct values have been set to h  5 km and α  0.5D∕L. Different control matrices are considered: the SVD method [Eq. (51), dashed line], MMSE NGS [Eq. (52), dash–dotted line], and MMSE LGS [Eq. (53), solid line].

Fig. 5. K-band SR as a function of LGS circle radius, for different numbers of LGSs, in cases of both high altitude (L  90 km, left) and low altitude (L  15 km, right) guide stars. The dash–dotted line displays the corresponding single LGS case, whereas the dashed line illustrates the single NGS case. The dotted vertical line shows the LGS angle corresponding to the edge of the telescope. Observations with 2, 3, and 6 LGSs are considered, as indicated on the plots.

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altitude and the angular location of the LGS. It shows that MMSE LGS estimator can tolerate large uncertainties of Δh ∼ 5 km and Δα ∼ 300 before reaching similar performance to that of the M ngs control matrix. The range even widens when compared to the standard SVD technique. These ranges depend neither on the true value of the turbulent layer height nor on that of the LGS angular location since the MMSE LGS curves will shift only along the x axis as a function of these values.

5. TOMOGRAPHY A. Compensating the Cone Effect with a Network of Guide Stars To circumvent the cone effect limitation, one can use a network of LGSs located at different positions in the sky and carrying out a tomographic reconstruction of the atmosphere. In the following, the LGSs will be radially distributed on a circle, the radius of which (so-called LGS field of view) can vary. Figure 5 shows the K-band SR as a function of the LGS field of view, for an increasing number of guide stars, for both cases of sodium (L  90 km) and Rayleigh (L  15 km) lasers. As expected, using several LGSs instead of one allows an increase in the quality of the correction. And the improvement is all the more significant when the single LGS is launched off-axis. For the sodium laser case, we can see that four LGSs are enough to fully cancel out the cone effect and reach the performance of an on-axis natural guide star. In contrast, the cone effect can only be partially compensated when using Rayleigh lasers, six LGSs allowing us to reach ∼70% of the K-band SR of the on-axis natural guide star. For the high altitude LGS system, the optimal LGS FOV is strongly marked and the performance can be severely degraded when the LGS circle deviates from this specific radius, especially when a few LGSs are used. The optimum is found to be for α ∼ 0.5D∕L, that is, when the circle that the LGSs draw on sky matches with the edge of the telescope aperture. This empirical law can also be deduced from rough geometrical considerations [7] noticing that α  0.5D∕L is the minimum angle that enables us to encompass the full volume of turbulence (the outer part of the LGS beams in that case being superimposed to that of the science star beam). Tokovinin et al. [34] have also found the same optimum from their numerical code (see, for example, Fig. 4 of their paper with an optimal radius of ∼900 for d  8 m, L  90 km, in the case

E. Tatulli and A. N. Ramaprakash

of 3LGS). The SR optimum is, however, not as sharp when the number of LGSs is larger than the number of turbulent layers. Also, the rule is valid only when the LGSs are significantly higher than the upper atmospheric layer, roughly when hupper ≤ L∕2. For low altitude LGSs such as Rayleigh stars, the situation is less clear. As the cone effect is stronger, the optimal angular radius will depend on the altitude of the upper turbulent layer. In that case, a theoretical analysis of the performance is suitable for a priori estimating the best radius of the LGS network according to the atmospheric properties of the observational site (most of all the altitude of the upper layer). B. Validity of Atmospheric Equivalent Layers Modeling The use of LTAO reconstruction requires the turbulent profile to be decomposed in discrete thin layers (so-called equivalent layers [45]), in order to achieve atmosphere tomography. We analyze in this section the validity of such a decomposition and estimate how many layers are needed to correctly describe the effects of a given continuous turbulent profile. We consider a classical Hufnagel continuous (night) profile [51], as shown in Fig. 6 (left), adjusting the parameters in order to obtain r 0  12 cm in the R band. We then slice this profile in N el equally thick zones and compute for each zone the height of the equivalent layer, such that the kth altitude verifies R hmax
k 2 h
k hC n
hdh hk  R min ; hmax
k 2 hmin
k C n
hdh

(56)

where hmin
k and hmax
k are the lower and upper limits of the kth C 2n zone. The turbulence associated to this layer is R hmax
k C 2n
hk  

hmin
k

C 2n
hdh

Δh

:

(57)

Figure 6 (middle and right) shows the evolution of the error as a function of the chosen number of equivalent layers N el . It is clear that, independent of the number and altitudes of the LGSs, the residual error quickly reaches a plateau, showing that only a few layers (∼4) are sufficient for a proper modeling of the LTAO correction. The plateau is less pronounced in the case of low altitude LGS, although the relative error on the phase residual estimate remains ≲5% when modeling

Fig. 6. Left: input Hufnagel C 2n
h profile. Middle: residual error as a function of the number of equivalent layers, for different numbers of laser spots at L  90 km, as indicated on the curves. Right: same as previously, but for L  15 km.

E. Tatulli and A. N. Ramaprakash

the atmosphere with four EL instead of 10. This translates into a relative error of ≲2% in the estimation of the associated K-band SR. Our study thus theoretically validates the relevance of the equivalent layer approach. It is consistent with Fusco analysis, which concluded that “only a small number of layers are needed to obtain a good precision on the statistical behavior of the turbulent phase” [44].

6. SODIUM VERSUS RAYLEIGH GUIDE STARS Sodium LGSs have been proven to provide better correction when compared to Rayleigh LGSs because of a much less severe cone effect. However, as sodium lasers are substantially more expensive by a factor of ≳10 and may even dominate the cost of the full AO system, it is interesting to compare the quality of AO correction between a single sodium LGS and several Rayleigh LGSs, as long as the overall cost of the AO remains smaller in the latter case. We investigate this

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trade-off for different classes of telescope diameter in the context of existing observatories where sodium laser devices have been installed and can be used for their AO system, namely the Lick (d  3 m, 7 × 7 subapertures), Gemini North (d  8 m, 12 × 12), VLT (d  8 m, 40 × 40), and Keck (d  8 m, 20 × 20) observatories. We have used experimental data obtained with combined MASS-DIMM site testing instruments [52] to estimate atmospheric conditions above the Mauna Kea Observatory (Gemini, Keck). It consists in six layers located at [0,1,2,4,8,16] km, with relative C 2n contributions of [53,11,4,12,9,11] %. For Mount Hamilton (Lick) and Paranal (VLT) observatories, we have used the theoretical Hufnagel night profile of Section 5.B. Both profiles have been generated for the same average seeing conditions (i.e., r 0  12 cm in the R band), although the Hufnagel C 2n is probably leading to more optimistic results as the contribution of the upper layers is lower in this case than that of the measurements at Mauna Kea.

Fig. 7. K-band performance of one sodium (solid lines) and three/six Rayleigh (dash–dotted/dashed lines) laser stars as a function of the photon flux. In the sodium case the star indicates the expected return flux for a 15 W laser. The filled star in the case of the VLT shows the current operating point according to Wizinowich [53]. For the Rayleigh case, the stars show the minimum power required for reaching the saturation regime, varying from 5 to 20 W.

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The numbers of SH subapertures of the WFSs correspond to the actual AO instruments in operation, as summarized by Wizinowich [53]. We have chosen an average AO bandwidth of 250 Hz and an associated loop closing time of τ  4 ms. Finally, we have set the detector noise to σ d  1e− . Figure 7 displays the K-band SR as a function of the incoming number of photons per subaperture, covering the different noise regimes as the photon flux increases: detector noise, photon noise, and fitting error/cone effect plateau, respectively. For the sodium LGS case, we have indicated the expected return photon flux of a 15 W laser with a star symbol. From the lidar equation [54], it corresponds roughly to a spot brightness of N ∼ 1.5 × 106 ph∕m2 ∕s or equivalently to a star magnitude of V ≃ 9.5. These numbers are consistent with the properties of the laser effectively used for Lick, Gemini, and Keck [53]. For the VLT, while the specification requires a return flux of N ≥ 1 × 106 ph∕m2 ∕s [55], it seems that the actual laser rather provides a V ≃ 11 artificial spot [53]. Both options (V ≃ 11 and V ≃ 9.5) are reported in this case. Alternatively, Rayleigh star return flux is indicated considering the minimum power required to reach optimal performance (saturation regime). We find that a minimum power of ∼5 W–20 W is needed, values that are mainstream numbers for that class of lasers [54]. We note that the contribution of LGS AO (i.e., without tip/tilt error) to the error budget obtained from Keck science images [56] (see Table 1 in their paper) gives a K-band SR of ∼0.5, which is in good agreement with our theoretical estimations (∼0.45). Similarly, the estimated (LGS/AO) K-band SR of ∼0.7 computed by Max et al. [57] for the Lick sodium LGS (see Table 1 in their paper) is consistent with our predictions (∼0.6). We, however, emphasize that the present sodium laser operating points are at the very edge or even below the plateau region that represents the maximum achievable performance. Although the estimation of the return flux is of debate since it will strongly depend on various factors, such as the sodium abundance in the mesospheric layer, we assert that more powerful lasers are quite likely to improve the performance of LGS AO correction of these observatories. The improvement would be significant especially for Lick and VLT telescopes with a potential K-band SR increase of ∼15%. It would, however, require lasers with power 2–5 times stronger than those in operation, hence driving to a substantial growth in the cost of the instrument. In the case of the Lick telescope, we can see that Rayleigh stars can represent a very interesting alternative since only three such lasers will allow reaching performance equivalent to that of the present sodium LGS. We therefore stress that, for the ≲5m class telescopes, this approach may offer excellent potential in terms of benefits/cost. On the contrary, the situation severely shifts in favor of sodium LGS when the telescope size increases, as the cone effect becomes too strong to be compensated by a network of several Rayleigh stars, as inferred by Le Louarn et al. [7]. Even doubling the number of Rayleigh stars from three to six is far from reaching the performance of sodium LGS. As a consequence, the Rayleigh LGS solution will not give the same performance as that of a sodium LGS for the ≳8 m class telescopes. However, if cost is a priority driver, for a loss in the K-band SR of ∼50%, one could get a multi-Rayleigh LGS at a fraction of the cost of a sodium LGS. In an even more drastic way, extremely

E. Tatulli and A. N. Ramaprakash

large telescopes (ELTs) with diameters of ≳30 m, which are expected to be built in the next decade, will be absolutely unable to work with Rayleigh laser guide systems.

7. CONCLUSION We have provided in this paper an analytical derivation of the performance of the LTAO technique, demonstrating that the phase residual error can be formally described by a combination of integrals of the product of three Bessel functions. Thanks to this formalism, we have quantified the limitations of AO performance arising from the combined effect of partial wavefront sensing, time delay, and the cone effect when using one LGS. The latter effect can be fought by using several guide stars and performing a tomographic reconstruction of the turbulent volume. In the case of sodium lasers, the compensation of focal anisoplanatism can be total with a moderate number (≥3) of artificial spots evenly distributed in the sky on a circle of angular radius 0.5D∕L. With Rayleigh stars, for which the cone effect is much stronger, focal anisoplanatism can be only partially corrected, even when using a great number of beacons, because the upper turbulent layers cannot be fully mapped by the laser beams. This fundamental limitation has often led us to consider Rayleigh stars unsuitable for astronomical purposes. However, when dealing with small diameter class telescopes (≲5 m), using a few (∼3) such lasers instead of a single sodium one should be considered a conceivable alternative, as it can provide equivalent AO correction with a lower overall cost of the instrument.

APPENDIX A: BESSEL AND ZERNIKE FUNCTIONS PROPERTIES 1. Integral Forms of Bessel Functions We recall the properties of the J m Bessel functions in their integral forms. They will be used to derive the equations of the following appendices: Z 2π cos
mγ exp
iy cos
γ − θk dγ 0



Z

2π 0

jmj

2π
−1 2 cos
mθk J jmj
y if m even ; jmj−1 2iπ
−1 2 cos
mθk J jmj
y if m odd

(A1)

sin
mγ exp
iy cos
γ − θk dγ



jmj

2π
−1 2 sin
mθk J jmj
y if m even : jmj−1 2iπ
−1 2 sin
mθk J jmj
y if m odd

(A2)

2. Zernike Polynomials Characteristics In polar coordinates. the Zernike modes are defined for a circular aperture without obstruction as Zm n
ρ; θ  Z j
ρ; θ

8 p p m < p2 cos
jmjθ if m > 0  n  1Rn
ρ 2 sin
jmjθ if m < 0 ; (A3) : 1 if m  0

where n and m are, respectively, the radial degree and the azimuthal frequency of the jth polynomial, j being defined as j 
n
n  2  m∕2, and

E. Tatulli and A. N. Ramaprakash

Rm n
ρ




n−jmj∕2 X s0

Vol. 30, No. 12 / December 2013 / J. Opt. Soc. Am. A


−1s
n − s! ρn−2s : s!
n  jmj∕2 − s!
n − jmj∕2 − s!

ˆ ks
κ as in Eq. (19), Assuming circular subapertures, that is, Π we obtain

(A4) The Zernike modes are orthonormal over a circle of unit radius, that is,

Z

2

Πp
ρZ j
ρZ k
ρd ρ 

1 if j  k 0 if j ≠ k

Z

Πp
ρZ j
ρΦ
Rρd2 ρ:

ϕj 

  Rs jκj Z J 2π 1 R iλRπ2π Rs exp−2iπρk ·κ d2 κ κx;y Qj
κ  jκj 2πAs R   Z J 1 2π RsRjκj iλ  κx;y Qj
κ exp2iπρk ·κ d2 κ: (A10) jκj Rs 

Dx;y kj

(A5)

with Πp
ρ being the unitary pupil function. For a given phase Φ
Rρ defined over a pupil of Pradius R, its Zernike decomposition is expressed as Φ
Rρ  ∞ j0 ϕj Z j
ρ, where the Zernike coefficients are calculated by projecting the phase on the polynomial basis:

Switching to polar coordinates with ρk  ρk ; θk  and κ  κ; γ, we have "

Dxkj

#

Dykj

(A6)

Qj
κ, the Fourier transform of Πp
ρZ j
ρ, can be written as p J
2πκ Qj
κ; γ 
−1n n  1 n1 πκ 8 p
n−jmj∕2 jmj > i 2 cos
jmjγ if m > 0
−1 > < p
n−jmj∕2 jmj ×
−1 i 2 sin
jmjγ if m < 0 : > > : n∕2 if m  0
−1

(A7)

3. Elements of the Interaction Matrix The elements of the interaction matrix D∞ are defined in Eq. (18) and can be rewritten as follows: λR 2πAs

Dx;y kj 

Z π

∂Πp
ρZ j
ρ k Πs ∂x; y

Dx;y kj 

λR 2πAs

ˆ ks
−κd2 κ: π · 2iπκx;y Qj
κΠ

Table 1. Z

2π 0

Z

0

0


 R dκJ 1 2π s κ J n1
2πκβxjmj−1;k J jmj−1
2πρk κ R

− βxjmj1;k J jmj1
2πρk κ;

(A12)

Dykj 

λ s Rs n;m

Z 0

∞


 R dκJ 1 2π s κ J n1
2πκβyjmj−1;k J jmj−1
2πρk κ R

βyjmj1;k J jmj1
2πρk κ;

(A13)

(A9) with

dγ cos
γ sin
jmjγexpiy cos
γ−θk   πsin
jmj − 1θk J jmj−1
y − sin
jmj  1θk J jmj1
y × 2π

0 2π

2π

∞

Evaluation of Integrals in Terms of Bessel Functions

Z

0

Z



0

Z

λ s Rs n;m

dγ cos
γ cos
jmjγexpiy cos
γ−θk   πcos
jmj − 1θk J jmj−1
y − cos
jmj  1θk J jmj1
y ×

2π

Z

Dxkj 

(A8)

By making use of the Fourier transform, the previous equation becomes Z


 Z iλ p ∞ R n1 dκJ 1 2π s κ J n1
2πκ πRs R 0 Z 2π  cos
γ  dγ exp2iπρk κ cos
γ−θk  × 0 sin
γ 8 p >
−1
n−jmj∕2 ijmj 2 cos
jmjγ if m > 0 > < p ×
−1
n−jmj∕2 ijmj 2 sin
jmjγ if m < 0 : (A11) > > : if m  0
−1n∕2


−1n

In the integral over γ, we recognize the Bessel functions of Appendix A.1 that we explicitly define in Table 1. This leads to the below expressions of the interaction matrix coefficients:




 R ρ d2 ρ: Rs

jmj−2

i
−1 2 jmj−1
−1 2 jmj−2

i
−1 2 jmj−1
−1 2

if jmjeven if jmjodd if jmjeven if jmjodd

dγ cos
γexpiy cos
γ−θk   2iπ cos
θk J 1
y

jmj

i
−1 2 if jmjeven jmj1
−1 2 if jmjodd

jmj i
−1 2 if jmjeven  −πcos
jmj − 1θk J jmj−1
y  cos
jmj  1θk J jmj1
y × jmj1 if jmjodd
−1 2 Z 2π dγ sin
γexpiy cos
γ−θk   2iπ sin
θk J 1
y

dγ sin
γ cos
jmjγexpiy cos
γ−θk   πsin
jmj − 1θk J jmj−1
y  sin
jmj  1θk J jmj1
y ×

dγ sin
γ sin
jmjγexpiy cos
γ−θk 

0

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J. Opt. Soc. Am. A / Vol. 30, No. 12 / December 2013

3n

sn;m  ijmj
−1 2 ( βxjmj1;k 

p n1

( p 2 1

if m ≠ 0 if m  0

cos
jmj  1θk  if m ≥ 0 sin
jmj  1θk 

if m ≤ 0

E. Tatulli and A. N. Ramaprakash

:


2
5 Z 0.023 λ D 3 L 5 dhζ
h3 C 2n
h R 5 2 ∞ 2 r0 π 23 0 C n
hdh Rs 0 2 Z∞ 
R 8 × dκ J 1 2π s κ κ −3 R 0 Z 2π pq pq × dγ cos2
γexp2iπρkl
hκ cos
γ−θkl
h ; (B6)

hsxk
αp sxl
αq i 

; (A14)

0

APPENDIX B: FORMAL DERIVATION OF Cov
s

pq where ρpq kl
h and θkl
h are the modulus and the argument of the vector ρl − ρk 
h∕Rζ
hΔαpq , respectively. For the integral over γ, once we rewrite cos2
γ as
1  cos
2γ∕2, we recognize the integral forms of Bessel functions of Table 1:

x;y 1. Computation of hsx;y k
αp sl
αq i Using Eq. (16), we have


hsxk
αp sxl
αq i   ×

λR 2πAs

2 ZZ

Z

subap
k;l

 ∂ ∂ Φlgs
Rρ1 ;αp  Φlgs
Rρ2 ;αq  d2 ρ1 d2 ρ2 : ∂x1 ∂x2 (B1)

Introducing the subaperture function Πs and the LGS covariance matrix of Eq. (7), we get  λR 2 2πAs

 ZZ R R × Πks ρ1 Πls ρ2 Rs Rs


hsxk
αp sxl
αq i 

×

pq

pq

dγ cos2
γexp2iπρkl
hκ cos
γ−θkl
h

pq pq  πJ 0
2πρpq kl
hκ − cos
2θ kl
hJ 2
2πρkl
hκ:

(B7)

We thus obtain the final expression for the moment hsxk
αp sxl
αq i as an integral of the product of Bessel functions:
2
5 Z 0.023 λ D 3 L 5 dhζ
h3 C 2n
h R∞ 2 R r π2 0 C n
hdh s 0 0 2 Z∞ 
Rs 8 × dκ J 1 2π κ κ−3 J 0
2πρpq kl
hκ R 0

hsxk
αp sxl
αq i 

∂2 Blgs
Rρ1 − ρ2 ; Δαqp d2 ρ1 d2 ρ2 ; (B2) ∂x1 ∂x2 Φ


2 2 ZZ

 λR R R 2 2 k l − d ρ 1 d ρ 2 Πs ρ Π ρ Rs 1 s Rs 2 2πAs ZL ∂2 BhΔn × ζ
h2
ζ
hRρ1 − ρ2   hΔαqp dh; ∂x1 ∂x2 0

0

2π

5 3

pq − cos
2θpq kl
hJ 2
2πρkl
hκ:

(B8)

The expression of associated moments hsyk
αp syl
αq i and hsxk
αp syl
αq i can be derived in a straightforward way by simple analogy:
2
5 0.023 λR D 3 R∞ 2 A r 2 0 C n
hdh s 0 Z ˆ ks
κΠ ˆ ls 
κκ2y jκj−113 × d2 κ Π ZL hΔαqp 5 × dhζ
h3 C 2n
hexp−2iπ Rζ
h ·κ

(B9)


2
5 0.023 λR D 3 R∞ 2 A r 2 0 C n
hdh s 0 Z ˆ ks
κΠ ˆ ls 
κκx κ y jκj−113 × d2 κΠ ZL hΔαqp 5 × dhζ
h3 C 2n
hexp−2iπ Rζ
h ·κ ;

(B10)

(B3) hsyk
αp syl
αq i 

where Δαqp  αp − αq . Further, we follow the analytical development of Molodij [58], with the intermediate change of variable η  ρ1 − ρ2 . In our case, however, we also take into account the derivative properties of the Fourier transform. Thus we have  Z λ 2 ˆ ks
κΠ ˆ ls 
κκ2x d2 κΠ As
 ZL hΔαqp 1 κ h exp−2iπ Rζ
h ·κ : × dh W Δn − 2 Rζ
h ζ
h 0 (B4)

5 3

0


hsxk
αp sxl
αq i 

hsxk
αp syl
αq i 

Using the definition of W hΔn
κ in Eq. (9), hsxk
αp sxl
αq i takes the generic form hsxk
αp sxl
αq i 


2
5 0.023 λR D 3 R∞ 2 A r 2 0 C n
hdh s 0 Z ˆ ks
κΠ ˆ ls 
κκ2x jκj−113 × d2 κ Π ZL hΔαqp 5 × dhζ
h3 C 2n
hexp−2iπ Rζ
h ·κ :

0

which, for circular subapertures, become

5 3

0

5 3


5 Z λ 2 D 3 L 5 dhζ
h3 C 2n
h 5R r0 π23 0∞ C 2n
hdh Rs 0 2 Z∞ 
Rs 8 × dκ J 1 2π κ κ−3 J 0
2πρpq kl
hκ R 0

hsyk
αp syl
αq i  (B5)

Switching to polar coordinates and assuming circular subapertures, we get

0.023




pq  cos
2θpq kl
hJ 2
2πρkl
hκ;

(B11)

E. Tatulli and A. N. Ramaprakash

Vol. 30, No. 12 / December 2013 / J. Opt. Soc. Am. A

hsxk
αp syl
αq i 


2
5 Z 0.023 λ D 3 L 5 dhζ
h3 C 2n
h R∞ 2 R r π2 0 C n
hdh 0 s 0 2 Z∞ 
R 8 pq × dκ J 1 2π s κ κ −3 − sin
2θpq kl
hJ 2
2πρkl
hκ: R 0

lgs hϕlgs 2
αp ϕ2
αq i 

5 3

2497


5 Z 4 × 0.023 D 3 L 5 dhζ
h3 C 2n
h R 5 ∞ 2 π23 0 C n
hdh r 0 0 Z∞ 14 × dκJ 2
2πκ2 κ − 3 J 0
2πρpq
hκ 0

 cos
2θpq
hJ 2
2πρpq
hκ;

(B18)

(B12) lgs hϕlgs 1
αp ϕ2
αq i lgs 2. Computation of hϕlgs 1;2
αp ϕ1;2
αq i From the definition of Zernike tip/tilt coefficients of Eq. (A6), we have




5 Z 4 × 0.023 D 3 L 5 dhζ
h3 C 2n
h R∞ 2 r π2 0 C n
hdh 0 0 Z∞ 14 × dκJ 2
2πκ2 κ− 3 − sin
2θpq
hJ 2
2πρpq
hκ: 5 3

0

lgs hϕlgs 1
αp ϕ1
αq i

ZZ



2 2 π p
ρ1 Z 1
ρ1 π p
ρ2 Z 1
ρ2 Blgs Φ
Rρ1 − ρ2 ; Δαqp d ρ1 d ρ2 :

(B13)

(B19)

lgs 3. Computation of hsx;y k
αp ϕ1;2
αq i Combining Eqs. (16) and (A6), we have

 ZZ
 λR R × Πks ρ1 π p
ρ2 Z 1
ρ2  2πAs Rs ∂ lgs B
Rρ1 − ρ2 ; Δαqp d2 ρ1 d2 ρ2 (B20) × ∂x1 Φ


Again, by my means of Fourier transform properties and variable changes of Molodij [58], the previous equation changes to Z 1 d2 κQ1
κQ1
κ R2
ZL × dhζ
h−2 W hΔn −

lgs hϕlgs 1
αp ϕ1
αq i 

0



 hΔαqp κ exp−2iπ Rζ
h ·κ Rζ
h (B14)


5 Z 0.023 D 3 11 d2 κQ1
κQ1
κjκj− 3 R 5 23 0∞ C 2n
hdh r 0 ZL hΔαqp 5 × dhζ
h3 C 2n
hexp−2iπ Rζ
h ·κ : 0

hsxk
αp ϕlgs 1
αq i 

 ZZ
 λR2 R d2 ρ1 d2 ρ2 Πks ρ1 π p
ρ2 Z 1
ρ2  Rs 2πAs ZL h ∂B × ζ
h Δn
ζ
hRρ1 − ρ2   hΔαqp dh; ∂x1 0




which in the Fourier plane rewrites in the following generic form: (B15)

hsxk
αp ϕlgs 1
αq i

From the expression of Q1 using Eq. (A7), we develop the equation in polar coordinates: lgs hϕlgs 1
αp ϕ1
αq i 

(B21)


5 Z 4 × 0.023 D 3 L 5 dhζ
h3 C 2n
h R 2 53 ∞ 2 π 2 0 C n
hdh r 0 0 Z∞ 14 × dκJ 2
2πκ2 κ− 3 0 Z 2π pq pq × dγ cos2
γexp2iπρ
hκ cos
γ−θ  ; 0

Z
λ ˆ ks
κQ
κκx d2 κΠ  −i 1 RAs
 ZL hΔαqp 1 κ h − exp−2iπ Rζ
h ·κ ; × dh W Δn 2 Rζ
h ζ
h 0 (B22)


5 Z λR D 3 ˆ ks
κQ
κκx jκj−113  −i 5 R ∞ 2 × d2 κΠ 1 23 0 C n
hdh As r 0 ZL hΔαqp 5 × dhζ
h3 C 2n
hexp−2iπ Rζ
h ·κ : (B23)


0.023

0

(B16) where ρ
h and θ are the modulus and the argument of
hΔαpq ∕Rζ
h. According to the integral definition of Bessel functions, we obtain pq

lgs hϕlgs 1
αp ϕ1
αq i

pq


5 Z 4 × 0.023 D 3 L 5  5 R∞ 2 dhζ
h3 C 2n
h r 3 π2 0 C n
hdh 0 0 Z∞ 14 × dκJ 2
2πκ2 κ− 3 J 0
2πρpq
hκ 0

− cos
2θpq
hJ 2
2πρpq
hκ:

(B17)

Similarly, from the definition of Q2 relative to the tilt Zernike coefficient ϕlgs 2 , we obtain

Using Eq. (A7) and assuming circular subapertures, the previous equation becomes in polar coordinates

5 Z 2×0.023 λ D 3 L 5 dhζ
h3 C 2n
h R 2 ∞ 2 R r π 2 0 C n
hdh s 0 0
 Z∞ R 11 × dκJ 1 2π s κ J 2
2πκκ− 3 R 0 Z 2π pq pq × dγcos2
γexp2iπρk
hκ cos
γ−θk  ; (B24)

hsxk
αp ϕlgs 1
αq i

5 3

0

pq where ρpq k
h and θk are the modulus and the argument of
hΔαpq ∕Rζ
h − ρk . Again we introduce the integral definition of Bessel functions so that we finally obtain

2498

J. Opt. Soc. Am. A / Vol. 30, No. 12 / December 2013



5 Z 2 × 0.023 λ D 3 L 5 dhζ
h3 C 2n
h R 5 ∞ 2 π23 0 C n
hdh Rs r 0 0
 Z∞ R 11 × dκJ 1 2π s κ J 2
2πκκ− 3 J 0
2πρpq k
hκ R 0

hsxk
αp ϕlgs 1
αq i 

pq − cos
2θpq k
hJ 2
2πρk
hκ:

E. Tatulli and A. N. Ramaprakash

Combining Eqs. (16) and (A6) leads to  ZZ
 λR R d2 ρ1 d2 ρ2 Πks ρ1 π p
ρ2 Z j
ρ2  2πAs Rs   ∂ lgs × ϕ
Rρ1 ; αp ϕ
Rρ2  ; (C1) ∂x1


hsxk
αp ϕj i 

(B25)

By analogy, we compute the remaining moments:

5 0.023 λR D 3 lgs y hsk
αp ϕ2
αq i  −i 5 R ∞ 2 23 0 C n
hdh As r 0 Z ˆ ks
κQ
κκ y jκj−113 × d2 κΠ 2 ZL hΔαqp 5 × dhζ
h3 C 2n
hexp−2iπ Rζ
h ·κ ; 0



5 0.023 λR D 3 R∞ 2 2 0 C n
hdh As r 0 Z ˆ ks
κQ
κκ x jκj−113 × d2 κΠ 2 ZL hΔαqp 5 × dhζ
h3 C 2n
hexp−2iπ Rζ
h ·κ ;

hsxk
αp ϕlgs 2
αq i  −i

hsyk
αp ϕlgs 1
αq i

(B26)

We perform the change of variable η  ζ
hρ1 − ρ2 [58], and we make use of the derivative properties of the Fourier transform to obtain the generic expression of the moment:

(B27)

hsxk
αp ϕj i Z
ZL λ ˆ ks 
ζ
hκQj
κκ x ζ
hW h d2 κ i dhΠ Δn RAs 0
 hαp κ exp2iπ R ·κ × R

5 0.023 λR D 3  i 5 R∞ 2 23 0 C n
hdh As r 0 Z ZL hα ˆ ks 
ζ
hκQj
κκx jκj−113 ζ
hC 2n
hexp2iπ Rp ·κ : × d2 κ dhΠ

5 3

0



5 λR D 3  −i 5 R ∞ 2 23 0 C n
hdh As r 0 Z ˆ ks
κQ
κκy jκj−113 × d2 κΠ 1 ZL hΔαqp 5 × dhζ
h3 C 2n
hexp−2iπ Rζ
h :κ ; 0.023

0

0

(C3) (B28)

which in polar coordinates and assuming circular subapertures gives

5 Z 2 ×0.023 λ D 3 L 5 dhζ
h3 C 2n
h R 5 ∞ 2 π23 0 C n
hdh Rs r 0 0
 Z∞ R 11 × dκJ 1 2π s κ J 2
2πκκ − 3 J 0
2πρpq k
hκ R 0

hsyk
αp ϕlgs 2
αq i 

pq cos
2θpq k
hJ 2
2πρk
hκ;

 ZZ
 λR2 R d2 ρ1 d2 ρ2 Πks ρ1 π p
ρ2 Z j
ρ2  Rs 2πAs ZL h ∂B × ζ
h Δn
Rζ
hρ1 − ρ2   hαp dh: (C2) ∂x1 0




(B29)

Using the definitions of Zernike polynomials and the circular subaperture Fourier transform, the previous equation can be rewritten in polar coordinates as follows:

5 Z 0.023 λ D 3 L dhC 2n
h R 2 ∞ 2 R r π 2 0 C n
hdh s 0 0
 Z∞ R 11 × dκJ 1 2πζ
h s κ J 2
2πκκ− 3 R 0 8 9 cos
jmjγ > > > > Z 2π < = p p × dγ cos
γ sin
jmjγ exp2iπρk
hκ cos
γ−θk
h ; > > 0 > > : ; 1

hsxk
αp ϕj ii
−1− 2 sn;m m

5 3

(C4) hsxk
αp ϕlgs 2
αq i


5 2 × 0.023 λ D 3  hsyk
αp ϕlgs
α i  R q 5 1 ∞ 2 π23 0 C n
hdh Rs r 0 ZL 5 × dhζ
h3 C 2n
h 0
 Z∞ R 11 pq × dκJ 1 2π s κ J 2
2πκκ − 3 − sin
2θpq k
hJ 2
2πρk
hκ: R 0


(B30)

where ρpk
h and θpk
h are the modulus and the argument of
hαp ∕R  ζ
hρk and sn;m is defined by Eq. (22). The different cases of the integral over γ are developed in Table 1. This finally leads to

5 Z 0.023 λ D 3 L dhC 2n
h R∞ 2 R r π2 0 C n
hdh 0 s 0
 Z∞ R 11 × dκJ 1 2πζ
h s κ J 2
2πκκ− 3 R 0

hsxk
αp ϕj i  sn;m

5 3

× βxjmj−1;k
θpk
hJ jmj−1
2πρpk
hκ

APPENDIX C: FORMAL DERIVATION OF Cov
s;ϕ 1. Computation of hsx;y k
αp ϕj i

− βxjmj1;k
θpk
hJ jmj1

2πρpk
hκ:

(C5)

Similarly, we obtain a generic expression in the y direction,

E. Tatulli and A. N. Ramaprakash

Vol. 30, No. 12 / December 2013 / J. Opt. Soc. Am. A

hsyk
αp ϕj i



5 0.023 λR D 3 R∞ 2 2 0 C n
hdh As r 0 Z ZL hα ˆ ks 
ζ
hκQj
κκy jκj−113 ζ
hC 2n
hexp2iπ Rp ·κ ; × d2 κ dhΠ

i

5 3

0

(C6)


5 Z λ D 3 L  sn;m 5 R ∞ 2 dhC 2n
h R r 3 π2 0 C n
hdh 0 s 0
 Z∞ Rs 11 × dκJ 1 2πζ
h κ J 2
2πκκ− 3 R 0 0.023




Z



0

dhQ1
ζ
hκQj
κW hΔn


 hαp κ exp2iπ R ·κ ; R (C9)


5 0.023 D 3 R 5 23 0∞ C 2n
hdh r 0 Z ZL hαp 11 × d2 κ dhQ1
ζ
hκQj
κjκj− 3 C 2n
hexp2iπ R ·κ : (C10) 0

Switching to polar coordinates with Eq. (A7), we get −2 hϕlgs 1 ϕj i  i
−1 sn;m m

Z

∞


5 Z 4 × 0.023 D 3 L dhζ
h−1 C 2n
h 5R π 2 23 0∞ C 2n
hdh r 0 0 11

dκJ 2
2πζ
hκJ n1
2πκκ− 3 8 9 cos
jmjγ > > > > Z 2π < = p p × dγ cos
γ sin
jmjγ exp2iπρ
hκ cos
γ−θ  ; > > 0 > > : ; 1

×

0

(C11) where ρp and θp
h are the modulus and the argument of
hαp ∕R. We use the results of Table 1 to finally derive


5 Z 2 × 0.023 D 3 L dhζ
h−1 C 2n
h R 5 ∞ 2 π23 0 C n
hdh r 0 0

∞

dκJ 2
2πζ
hκJ n1
2πκκ 0 × βyjmj−1;k
θp J jmj−1
2πρp
hκ

×

which in the Fourier plane becomes L


5 0.023 D 3 R∞ 2 r 2 0 C n
hdh 0 Z ZL hαp 11 × d2 κ dhQ2
ζ
hκQj
κjκj− 3 C 2n
hexp2iπ R ·κ ; 5 3

 sn;m

0

Z

(C12)

(C7)

hϕlgs 1 ϕj i ZZ  d2 ρ1 d2 ρ2 π p
ρ1 Z 1
ρ1 π p
ρ2 Z j
ρ2 hϕlgs
Rρ1 ; αp ϕ
Rρ2 i ZZ  d2 ρ1 d2 ρ2 π p
ρ1 Z 1
ρ1 π p
ρ2 Z j
ρ2  ZL dhBhΔn
Rζ
hρ1 − ρ2   hαp ; (C8) ×

d2 κ

− βxjmj1;k
θp J jmj1

2πρp
hκ:

0

2. Computation of hϕlgs 1;2
αp ϕj i From Eq. (A6), we have

Z

0

14

dκJ 2
2πζ
hκJ n1
2πκκ − 3

(C13)

 βyjmj1;k
θpk
hJ jmj1

2πρpk
hκ:

1 R2

∞

× βxjmj−1;k
θp J jmj−1
2πρp
hκ

hϕlgs 2 ϕj i 

× βyjmj−1;k
θpk
hJ jmj−1
2πρpk
hκ

hϕlgs 1 ϕj i 

Z ×


5 Z 2 × 0.023 D 3 L dhζ
h−1 C 2n
h R 5 ∞ 2 π23 0 C n
hdh r 0 0

The moment associated to the tilt coefficient is deduced from above by straightforward analogy:

which assuming circular subapertures changes to hsyk
αp ϕj i

hϕlgs 1 ϕj i  sn;m

2499

−14 3

 βxjmj1;k
θp J jmj1

2πρp
hκ:

(C14)

ACKNOWLEDGMENTS We thank Dr. Warren Skidmore for providing the experimental turbulent profiles of the Mauna Kea observatory and for his useful comments regarding the methods used to obtain these measurements. The authors are grateful to the anonymous referee, whose careful review of the paper helped to improve its clarity and quality.

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