A tomographic algorithm to determine tip-tilt information from laser guide stars

0.87 MB·

by  A. P. Reeves

MNRAS000,1–9(2015) Preprint16March2016 CompiledusingMNRASLATEXstylefilev3.0
A tomographic algorithm to determine tip-tilt information
from laser guide stars
A. P. Reeves,1(cid:63) T. J. Morris,1 R. M. Myers,1 N. A. Bharmal1 and J. Osborn1.
1Centre for Advanced Instrumentation, Durham University, Durham, DH1 3LE, UK
6
1 AcceptedXXX.ReceivedYYY;inoriginalformZZZ
0
2

r ABSTRACT
a Laser Guide Stars (LGS) have greatly increased the sky-coverage of Adaptive Optics
M
(AO) systems. Due to the up-link turbulence experienced by LGSs, a Natural Guide
Star (NGS) is still required, preventing full sky-coverage. We present a method of ob-

5 taining partial tip-tilt information from LGSs alone in multi-LGS tomographic LGS
1
AOsystems.ThemethodofLGSup-linktip-tiltdeterminationisderivedusingageo-

metricapproach,thenanalterationtotheLearnandApplyalgorithmfortomographic
]
M AO is made to accommodate up-link tip-tilt. Simulation results are presented, verify-
ingthatthetechniqueshowsgoodperformanceincorrectinghighaltitudetip-tilt,but
I
notthatfromlowaltitudes.Wesuggestthatthemethodiscombinedwithmultiplefar
.
h off-axis tip-tilt NGSs to provide gains in performance and sky-coverage over current
p
tomographic AO systems.
-
o
Key words: adaptive optics – atmospheric effects
r
t
s
a
[

2 1 INTRODUCTION These AO configurations use information from a number of
v LGSs, off-axis from the science target, to estimate the sci-
The use of Laser Guide Stars (LGSs) in Adaptive Optics
9 ence path turbulence. Such systems overcome the so called
(AO) has greatly increased the area of the sky available for
1 cone-effect where the LGS samples a cone of turbulence in
correction, from ≈10% up to ≈85% (?). In turn this has
8 the science path rather than the full cylinder of turbulence
7 led to a vast increase in the number of astronomical sci- seen by light from the science target (?). They can also
0 ence targets which can be observed using AO. The laser
achievealargecorrectedfieldofviewinthecaseofMCAO,
. experiences turbulence whilst traveling upwards to form an
1 oralarge‘fieldofregard’inthecaseofMOAO.Tomographic
artificial guide star, so its position will move in the sky. It
0 LGSsystemsstillrequireaNGStoestimatetip-tiltmodes,
is thought that this effect renders all ‘tip-tilt’ information
6 limiting their potential sky coverage. Suitable NGS are no-
1 gained from LGS Wave-Front Sensors (WFS) useless, as it toriously absent from much of the sky around the galactic
: is a function of LGS ‘up-link’ movement and the desired
v poles,wheremanyscientificallyinterestingtargetsexist(?).
‘down-link’ tip-tilt, which have previously been considered
i
X tobeentangledirretrievably.Ithasevenbeensuggestedthat
the tip-tilt the laser acquires on the up-link is the recipro- MethodstoobtainallcorrectioninformationfromLGSs
r
a cal of the global tip-tilt on the down-link path, hence little alone have been proposed. Some have not yet been imple-
tip-tiltwillbeobservedontheWFSatall(?).Tocorrectly mented due to the requirement of complex laser schemes
obtainthesciencepathtip-tilt,aNaturalGuideStar(NGS) and/or auxiliary beam viewing telescopes (??????). ? dis-
must still be used (?). As a tip-tilt WFS requires relatively cussestheuseofthetemporaldelaybetweenthelaunchtime
few photons and the anisoplanatic patch size is large for of a laser pulse and the time it is received by the telescope
tip-tilt modes, the requirements on a NGS are much less- to estimate up-link tip-tilt. This technique requires the use
ened (?). Nonetheless, requiring a tip-tilt NGS still limits ofthefulltelescopeapertureforlaserlaunch,whichimplies
the sky-coverage of an LGS AO system. some problems with scattered light and fluorescence. More
Tomographic LGS configurations, such as Laser Tomo- recently,?hasproposedanLGSassistedluckyimagingsys-
graphic AO (LTAO)(?), Multi-Object AO (MOAO) (??) tem, which could provide full sky AO coverage but entails
andMulti-ConjugateAO(MCAO)(??),arecomingonline. discarding some science flux and would not be suitable for
spectroscopy.?exploredthepotentialusageofLGSAOwith
no tip-tilt signal, allowing 100% sky-coverage. It was found
(cid:63) E-mail:[email protected](APR); thatforsomeapplications,adedicatedNGStip-tiltstarwas
(cid:13)c 2015TheAuthors
2 A. P. Reeves, T.J.Morris, R. M. Myers, N. A. Bharmal and J. Osborn
not required and a telescopes fast guiding system was ade-
quate. 1.0
In this paper we propose a method to retrieve partial Theoretical
tip-tilt information from a number of LGSs in existing or Simulated
currently proposed tomographic AO systems from only the 0.8
systems wavefront sensor measurements. This is similar to
that proposed by ?, though we do not required the knowl-
edge of the exact sky position of one of the LGS. We aim n0.6
o
tpoerifmorpmroevdewAitOhpnoertfoipr-mtialtncNeGaSc,ropsostethnteiawllhyorleelaskxyingovtehreArOe- orrelati
C0.4
quirementfor,orforsomeapplicationseliminatingtheneed
for, a NGS.
If the tip-tilt modes measured across the full-aperture 0.2
are the same as that across the beam launch aperture, it
isclearthatthetip-tiltsignalwouldindeedbeirretrievable
from LGS WFSs, as they would be reciprocal and little at- 0.0
0.0 0.2 0.4 0.6 0.8 1.0
mospheric tip-tilt would be observed from a LGS WFS at DLLT/D
all. In § 2, we show that the tip-tilt modes across the beam
launchtelescopeareuncorrelatedwiththoseoverthewhole
aperture, opening the possibility of LGS up-link tip-tilt de- Figure 1. Theoretical and simulated correlation of phase per-
turbations in Kolmogorov turbulence between concentric tip-tilt
termination.Thealgorithmforup-linktip-tiltdetermination
isderivedin§3andanadaptationtotheLAalgorithmpro- modes as a function LLT diameter, DLLT, as a fraction of the
telescopediameter,D.
posedby?forMOAOissuggestedasapracticalmethodfor
its use. Results from simulation verifying the technique are
presentedin§5.Finally,in§6wediscussthepracticaluses
Zernike polynomials, R is the radius of the telescope, J
n+1
of an LGS up-link retrieving AO system, and how it may
andJ areBesselfunctionsofthefirstkind,kisthewave
p+1
provide increases in LTAO sky-coverage and performance
numberofthelightandr istheatmosphericFriedparam-
0
by combination with ground layer tip-tilt correction.
eter (?).
Thecovariancebetweenconcentrictip-tiltmodesofdif-
ferent radii are plotted in Figure 1 (where n,p =1). A plot
of the correlation of tip-tilt modes in ten thousand simu-
2 CORRELATION OF TIP-TILT BETWEEN
latedrandomKolmogorovphasescreensisalsoplotted.Itis
TELESCOPE AND BEAM-LAUNCH
evident that the correlation of tip-tilt modes between small
APERTURES
and large apertures in the regime where D /D < 0.1 is
LLT
Iftheglobaltip-tiltacrossthetelescopeapertureisidentical lessthan0.1.Thisistrueforboththetheoreticalexpression
tothatoverthebeamlaunchtelescope,anytip-tiltencoun- andthesimulatedphasescreens,whichmatchcloselyinsuch
teredbytheLGSup-linkpathwillhaveanequalbutoppo- a regime. This result means that tip-tilt modes will not be
site effect on the return path (though with a slight change reciprocal and will be visible on an LGS WFS. Observed
due to temporal delay (?)). Consequently little tip-tilt will tip-tilt will be some function of the turbulence encountered
be observed on the LGS WFS and the tip-tilt component by the laser as it propagates up to form an artificial guide
of the science path can not be determined by that WFS. star and the global tip-tilt across the telescope aperture as
This is referred to as tip-tilt ‘reciprocity’ and is the case it propagates back. It should be noted that the tip-tilt ob-
if the laser is launched from the full aperture of the tele- served by the laser will be larger than that seen by the full
scope. All current facility LGS AO systems use a separate aperture due to greater spatial averaging on larger scales.
Laser beam Launch Telescope (LLT). On these telescopes Despitebeinguncorrelated,itispossiblethatbychance
andthoseplannedforthefuture,DLLT <<D,whereDLLT thereisasimilarcomponentoftip-tiltacrossboththelaunch
denotes the diameter of the LLT and D is the size of the andtelescopeapertures,inwhichcaseitwillnotbeobserved
telescope aperture. onaLGSWFS.IfmultipleLGSsareused,thenthechance
DeterminationofLGSup-linktip-tiltcanonlybepossi- of the same component of tip-tilt being present across all
bleiftheup-linkanddown-linktip-tiltcomponentsareun- launch paths is significantly reduced.
correlated or it will not be fully observed by the WFS. The
covariance between two concentric Zernike modes of differ-
entradiiinKolmogorovturbulenceisshowninequation(1)
(?), 3 TOMOGRAPHIC LGS TIP-TILT
DETERMINATION
C=0.0145786e12iπ(n−p)(cid:112)(n+1)(p+1)(cid:18)R(cid:19)5/3 3.1 Retrieving down-link turbulence induced
r0 slopes
(cid:90) ∞ J (2πk)J (2πγk)
× dk n+1 p+1 , (1) As demonstrated in the previous section, the measurement
γk14/3
0 from a LGS WFS is a function of the atmospheric turbu-
where γ represents the fractional size relationship between lence the laser propagates through on the way up to form
the two apertures, n and p are the radial orders of the two a guide star and the turbulence the return light propagates
MNRAS000,1–9(2015)
Tomographic laser guide star tip-tilt determination 3
through as it travels back down to the telescope. For AO Actual position
correctionofanastronomicalsciencetargetthetwocompo- Expected of LGS αdue to
nents must be separated and only the latter is required. If position of α uplink β
LGS α
using a Shack-Hartmann or Pyramid WFS, WFS measure- turbulence
mentswillbeintheformofslopesrepresentingthegradients
of the measured phase within any given sub-aperture. The
useofsuchagradientmeasuringWFSisassumedinthefol-
lowing derivations. We can express slopes measured by an
LGS WFS as the sum of the laser up-link induced slopes
and the down-link turbulence induced slopes,
˜s=˜s +˜s , (2)
l t
where ˜s is a vector representing the slopes measured
on a WFS,˜s is a vector representing the slopes caused by
l
LGS up-link turbulence and˜s is a vector representing the
t
slopes caused by down-link turbulence. For AO correction H
of a natural astronomical science target we must obtain˜s .
t
NotethatLGSup-linkturbulenceresultsexclusivelyintip-
tilt modes being observed on the WFS and no higher order
modes,so˜s willbehomogeneousinthexandy directions.
l
For an AO system with a single LGS and no external refer-
ence, determining˜s is not possible as there is not enough Turbulence
t
layer at
informationtodetermine˜s.Inatomographicsystem,there
l height, h
is more information about the turbulence sampled by the
LGSs on the up-link, and˜s can be computed.
t
For the remainder of this section we consider a trivial
h
2-dimensional, tomographic, two LGS AO geometry, where
bothLGSsarecentre-launched.Thefollowingapproachcan
bescaledtomanycentrelaunchedLGSs,thoughthemath-
ematicsquicklybecomescumbersomewithmorethanthree.
TheLGSsarelabeledLGSαandLGSβ andtheobserving Sub-aperture: 1 … n
WFSs as WFS α and WFS β. Slopes measured on WFSs
are denoted as˜s and˜s respectively. This geometry is il-
α β
lustrated in Figure 2. Figure2.ThegeometryoftheLGSsystemunderconsideration.
WFS β observes the area of turbulence which causes One turbulent layer is shown, which features only a tilt at the
the up-link tilt on WFS α, hence we postulate that there pointthatLGSαoverlapswiththefieldofviewofsub-aperture
is a transform Tˆ which relates the down-link turbulence nonWFSβ.
αβ
induced slopes,˜s , to up-link induced tip-tilt measured on
βt
WFS α,˜s ,
αl and substituting into equation (4),
˜sαl =Tˆαβ˜sβt. (3) Tˆαβ˜sβ =Tˆ−βα1(Tˆβα˜sα−˜sβl)+Tˆαβ˜sβl
We initially consider the simple situation illustrated in =˜sα+(Tˆαβ−Tˆ−βα1)˜sβl. (7)
Figure2,whereasingleturbulentlayerataheighth,which Finally, re-arranging for˜s ,
βl
featuresonlyatiltinthesectionwhereLGSαoverlapswith
˜s =(Tˆ −Tˆ−1)−1(Tˆ ˜s −˜s ) (8)
the field of view (FOV) of sub-aperture n on the WFS ob- βl αβ βα αβ β α
servingLGSα.Inthiscaseitisclearthatsuchatransform, and similarly for˜s
αl
Tˆ ,existsandcanbetriviallycomputedasWFSβisunaf-
fecαtβedbyup-linkturbulenceso˜s =0,hence˜s =Tˆ ˜s . ˜sαl =(Tˆβα−Tˆ−αβ1)−1(Tˆβα˜sα−˜sβ). (9)
βl αl αβ β
Ingeneralhowever˜sβtwillnotbeknown,asLGSβwillalso ˜sα and˜sβ aretheWFSmeasurementsandtheTˆ trans-
experience up-link tip-tilt. For this general case, forms can be obtained by considering the geometry of the
system i.e., where sub-apertures from a WFS observe the
Tˆ ˜s =Tˆ (˜s +˜s )
αβ β αβ βt βl up-link path of the other laser(s). It is now possible to cal-
Tˆ ˜s =˜s +Tˆ ˜s (4) culate the turbulence induced slopes, as˜s =˜s−˜s. These
αβ β αl αβ βl t l
aretheslopeswhichwouldhavebeenmeasuredfromaguide
and star with no up-link tip-tilt effects, and can be used to per-
form the AO reconstruction without the requirement of an
Tˆ ˜s =˜s +Tˆ ˜s . (5)
βα α βl βα αl NGS for tip-tilt measurement. The above analysis can be
We now have 2 equations in (4) and (5), to solve for 2 performed for more complex LGS AO systems with many
unknowns,˜s and˜s . Re-arranging equation (5), LGSs in other geometries.
αl βl
In general there will be more than one discrete turbu-
˜s =Tˆ−1(Tˆ ˜s −˜s ) (6) lent layer in the atmosphere, hence the measurement of a
αl βα βα α βl
MNRAS000,1–9(2015)
4 A. P. Reeves, T.J.Morris, R. M. Myers, N. A. Bharmal and J. Osborn
particular element in ˜s which overlaps with LGS α will each direction, Tˆ is
βt αβ
not just represent the turbulence at height h, but will be
the sum of measurements from all turbulent layers. This H−Hh1 H−Hh2 ··· H−HhN
risepmreitsiegnattsedsombyeinnocirseeaisnintghethmeenausumrbemereonftLofG˜sSαsl,. Tsuhcehntohiaset Tˆαβ = 2λπ H−H...h1 H−H...h2 ·.·.·. H−H...hN, (11)
other layers from non-overlapping heights average to zero,
H−h1 H−h2 ··· H−hN
leaving only the common measurement of the slope at the H H H
point LGS α overlaps with the layer at altitude h. where h denotes the centre of the vertical height ‘bin’ re-
n
For the centre launched case, the slopes due to down- solvable by the sub-aperture n. By considering the system
link turbulence, st, cannot be determined for a turbulent geometry, including the launch angle of LGS α and β, θα
layer at the ground. For a layer at this height, ˜sαt = ˜sβt, andθβ respectively,weshowinAppendixBthathn canbe
˜sαl = ˜sβl, ˜sα = ˜sβ, and there is no-longer more than one expressed as
independent equation from which to determine˜s and˜s .
αl βl H(D −(n+0.5)d)
Further to this issue, if the lasers are centre launched, then h = 2 . (12)
the beam paths are likely to be obscured by the“shadow” n D2 −(n+0.5)d+H(θα+θβ)
of the secondary obscuration, the turbulence that the laser
Again, this relationship can be extended for any number of
passes near the ground layer is not measured.
LGS WFSs, where WFS β observes the path of LGS α.
AnAOsystemthatlaunchesLGSsfromdifferentpoints
The final step in creating an LGS up-link transform
within the telescope aperture could potentially overcome
is to tailor the matrix to the required atmospheric turbu-
theselimitations.Inthiscase˜s (cid:54)=˜s atthegroundlayer,
αl βl lence vertical profile. Each column in the matrix shown in
so both can be determined. Depending on the laser launch
equation (11) represents a vertical height bin resolvable by
scheme,itisalsopossiblethatthelowlayerturbulenceinthe
the tomographic LGS AO system. If a turbulence profile is
beampathcouldbeobserved.Suchasschemedoeshowever
known, then columns that represent heights where there is
entail other difficulties, such as scattering from launching
negligibleturbulencecanbesettozero.Thisstepwillreduce
the laser directly off the primary mirror.
thenoisecontributedby‘falselayers’whichcouldotherwise
A system with LGSs launched from outside the tele-
be detected, where random perturbations from real turbu-
scope aperture (side launched) is unlikely to be suitable for
lent layers or noise could seem like turbulence at a height
this method of LGS up-link tip-tilt correction as a LGS’s
where no turbulence is present. Such a profile can be ob-
launchpathisnotobservedbyotherLGSWFSs.Itispossi-
tained from either the tomographic AO system itself or an
blethatouterWFSsub-aperturescouldbeusedastheymay
external profiling instrument.
correlate strongly with the launch path turbulence, though
this is outside the scope of this work.
4 A LEARN AND APPLY APPROACH
The geometric approach described in the previous sections
3.2 Obtaining LGS up-link transforms toestimateandrecoverLGStip-tiltmodesisclearlyhighly
idealised. It requires knowledge of the turbulence C2 verti-
n
The LGS up-link matrices describing the response of LGS
cal profile and that the calibration of the LGS WFSs and
motion to WFS measurements are defined in equation (3),
pointing of the LGSs is perfect. It would also not take into
they relate down-link turbulence measurements of a WFS
accountourunderstandingofatmosphericturbulencestatis-
to the predicted up-link path of another LGS. They can be
tics to improve correction. As correlation between adjacent
calculatedbyconsideringthegeometryillustratedinFigure
sub-apertures can be significant (?), information from sub-
2andtheeffectofalayerataheighthwithasmallregionof
apertures around those which view the up-link path of an-
turbulence where the FOV of sub-aperture n overlaps with
other LGS can be used to improve estimation of its up-link
LGS beam α, at height H.
tip-tilt. This also allows an estimate to be made of ground
Foragivensub-aperture,sαl istheslopemeasureddue layer tip-tilt, even for centre launched LGS AO systems,
toup-linktip-tiltonWFSα.Itisrelatedtotheslopemea- from sub-apertures surrounding the central obscuration.
sured on a corresponding sub-aperture on WFS β which LearnandApply(LA)isamethodusedintomographic
views LGS α at height h, sβt. It is shown in Appendix A AO systems, such as MOAO and LTAO, for open-loop to-
that mographic reconstruction which accounts for atmospheric
turbulence statistics and the calibration of an AO system
λ H−h
s = s . (10) (?).InsteadofusingapurelygeometricalapproachforLGS
αl 2π H βt
up-link tip-tilt determination, LA can be modified to im-
plicitly account for up-link tip-tilt. LA is briefly described
Thesystemhasonlyafiniteverticalresolution(defined
below.
by the number of sub-apertures and LGS asterism separa-
If there is a linear relationship between off-axis WFS
tion)tocorrectforturbulentlayers.Sub-aperturenwillob-
measurements,˜s , and WFS measurements on-axis to the
servetheintegratedturbulencebetweenwheretheLGSen- off
direction of a science target,˜s , one can write
tersitsFOVandwhereisexitsitsFOV.Differentgroupsof on
sub-apertures on a WFS will correspond to different turbu- ˜s =Wˆ.˜s (13)
on off
lent layer heights for which they can predict up-link tip-tilt
on other LGS WFSs. Recalling that s is homogeneous in where Wˆ is the tomographic reconstructor. If Wˆ can be
αl
MNRAS000,1–9(2015)
Tomographic laser guide star tip-tilt determination 5
obtained,itcanbeusedtocalculatepseudoWFSmeasure- equation (2),
ments in the direction of a potential science target, which
(cid:104)s s (cid:105)=(cid:104)(s +s )(s +s )(cid:105)
can then be used to calculate DM commands to provide α β αt αl βt βl
correction in that direction. To estimate it, a large number =(cid:104)sαtsβt(cid:105)+(cid:104)sαtsβl(cid:105)+(cid:104)sαlsβt(cid:105)+(cid:104)sαlsβl(cid:105). (16)
of uncorrected WFS measurements of both on and off-axis
Oftheseterms,thefirstisonlyaresultofdown-linkturbu-
slopes may be taken. In this case, the set of on-axis mea-
lenceandisthesameasthecovariancematrixwhichwould
surements are denoted as Mˆ and the set of off-axis mea-
on berequiredinconventionalLearnandApply.Thistermcan
surements as Mˆ . Vidal et al. show that a tomographic
off becalculatedinaformsimilartothatwhich?usetoobtain
reconstructor for these specific measurements, Wˆ(cid:48), can be
thecovariancematricesbetweenseparatedNGSWFSmea-
expressed as
surements with some modification to account for the cone
Wˆ(cid:48) =(Mˆ Mˆt )(Mˆ Mˆt )−1. (14) effect associated with LGSs.
on off off off
The second and third terms describe the relationship
IfthenumberofmeasurementstakentoformMˆ and
on betweentheobserveddown-linkturbulenceandthetipand
Mˆ approachesinfinity,thetomographicreconstructorWˆ(cid:48)
off tilt observed by another WFSs due to the patch of turbu-
approaches Wˆ, a general reconstructor which can recon-
lence that the lasers pass through on their up-link paths.
structanysetofslopesforthegivenguidestargeometryand
They can be calculated again by considering the separation
atmosphericturbulenceprofile.Inthislimit,theexpressions
ofeachsub-apertureonthedown-linkwiththelaunchpath
Mˆ Mˆt andMˆ Mˆt approachCˆ andCˆ ,the
on off off off OnOff OffOff foreachlaser.Astheyareformedbyalargetiportiltfrom
covariancematricesbetweenon-axisandoff-axisslopes,and
one WFS correlating with measurements from a single, or
off-axis and off-axis slopes respectively. The generalised to-
small number of, sub-aperture(s) from another WFS, it is
mographic reconstructor may now be expressed as
expected that they will appear as a matrix of vertical and
Wˆ =Cˆ ×Cˆ−1 . (15) horizontal stripes.
OnOff OffOff
The final term is the covariance between the up-link
If the profile of the atmosphere and system calibration
inducedtip-tiltmeasurements.Thisvalueisdependentupon
is well known, both covariance matrices can be calculated
the separation between the two laser paths at an altitude
purely analytically from statistical descriptions of turbu-
layer and as it is a result of only tip and tilt, it is constant
lence.Asthissituationisnotoftenthecase,analternativeis
foreachpairofWFSs.Assumingacentrelaunchedcase,this
torecordsomedatafromthesysteminopenlooptocreatea
term will be large for low altitude layers, where the up-link
‘raw’ covariance matrix which contains information regard-
laser paths overlap and small at high layers where the laser
ingtheatmosphericprofileandAOsystemcalibration.The
pathsareseparated.Asitisconstant,thisvaluereducesthe
‘raw’ covariance matrix cannot be used alone to create the
contrast of the the covariance matrices and so effectively
tomographicreconstructorasitisnotgeneralisedandwould
make them less useful. Hence, it is again expected that this
also contain errors due to noise. It can though be used to
approach will work well at higher layers where this term is
actasareferencetofitananalyticallygeneratedcovariance
small, but less well for low layers where it will dominate.
matrixwhichisgeneralisedandnotpronetonoiseeffects.In
Thesimulationsperformedin§5useaLAapproachto
thiswayLAcreatesageneralisedtomographicreconstructor
predictLGSup-linktip-tilt.Wedonotyetattempttoderive
which accounts for AO system calibration and our statisti-
the analytical form of the required covariance matrices. We
cal description of atmospheric turbulence. This process is
instead rely on the fact that we can simulate a very large
termed the ‘learn’ stage of the reconstructor.
number of uncorrelated phase screens to create a large set
Oncebothcovariancematriceshavebeencomputed,the
of‘learn’slopestocomputecovariancematricesbetweenoff-
reconstructor, Wˆ, can be formed and ‘applied’ to off-axis
axis and on-axis slopes. Deriving the analytical covariance
slopes to give an estimation of on-axis slopes. The LA al-
matrices could improve on the performance we show and
gorithm hasbeentestedsuccessfully bothinthe laboratory
would be essential for use on-sky.
and on-sky by the CANARY MOAO demonstrator (?).
We propose that the LA algorithm is also applicable
for LGS tip-tilt determination, as it was demonstrated in
§ 3 that the required science direction slopes are a linear 5 SIMULATION RESULTS
functionoftheoff-axisLGSmeasurements.Theadvantages
5.1 Simulation set-up
of using LA are many fold. LGS tip-tilt determination can
accountforsystemalignmentandLGSpointing.Themath- Inthefollowingsimulationsweusethemodified,tip-tiltre-
ematicsshownin§3doesnothavetoberepeatedforhigher trieval LA based algorithm to perform LGS AO correction
numbers of LGS, which quickly becomes cumbersome. The withnoNGStip-tiltWFSandcomparetheseresultstothe
turbulenceprofiledoesnothavetobeexternallymeasuredto currently used LTAO configuration, where tip-tilt informa-
averyhighverticalresolution.Finallyandperhapsmostim- tionisremovedfromLGSslopes,andalow-orderNGSused
portantly, the use of covariance matrices implicitly includes togettip-tiltinformation.Weshowa“best-case”scenariofor
information about LGS up-link from sub-apertures near to LTAO, where the tip-tilt NGS is on-axis. To show that the
those identified as geometrically observing a LGS beam. tip-tilt determination is working correctly, we also simulate
TouseLA,itmustbealteredtoaccountforthefactthat anLGStomographicsystemwhereLGStip-tiltinformation
thetip-tiltsignalfromLGSWFSsisnolongerremoved.The is removed but no NGS is used for tip-tilt correction. In all
analyticalformofslopecovariancematricesinthiscasemust simulatedAOmodesthelasersarelaunchedfromthecentre
be derived. We consider the covariance between two WFS ofthetelescopeaperture,behindthesecondaryobscuration.
separated slope measurements with the definition given in The code used to perform these simulations has been
MNRAS000,1–9(2015)
6 A. P. Reeves, T.J.Morris, R. M. Myers, N. A. Bharmal and J. Osborn
Ttilatbrleetr1ie.vPalarametersusedinthesimulationofLGSup-linktip- hsαtsβti hsαlsβli
s
Parameter Value e
p
o
TCeelnetsrcaolpoebpscriumraatriyondidaimameteetrer(m(m)) 81.1 βy-sl
S
Phasepointsinpupil 256×256 WF
Totalphasescreensize(m) 128×128
s
Radiusofoff-axisLGSasterism(arc-seconds) 10 e
p
WFSsub-apertures 16×16 slo
DMActuators 17×17 x-
Sciencefieldwavelength(µm) 1.65
Scienceintegrationtime(s) 60
Loopframerate(Hz) 400 hsαtsβli hsαlsβti
s
e
p
writtensolelyinthePythonprogramminglanguageandin- o
sl
corporates full physical optical propagation of LGS as they y-
β
pass up through the atmosphere to form a guide star. This S
F
isnecessarytoaccuratelyestimatetheLGSup-linkpathas W
the Fresnel number for a typical LGS beam is ≈ 1 , mean- es
p
ing diffraction cannot be ignored (?). A geometric ray trac- slo
ing method is used to calculate the wavefront measured on x-
theWFSfromturbulenceencounteredaslightpassesdown
x-slopes y-slopes x-slopes y-slopes
through the atmosphere. The PSF formed by the LGS up-
WFSα WFSα
link is then convolved with each sub-aperture PSF to give
realistic simulation of LGS up-link turbulence. Focal aniso-
planatism (cone effect) is included when propagating light Figure 3. A simulated LGS up-link covariance matrix between
two LGS WFSs observing in different directions deconstructed
downfromanLGS.Thesesimulationsdonotincludeeither
into its constituent terms. The top-left image shows the covari-
read noise or photon shot noise. Simulation parameters are
ance of down-link turbulence. The bottom two images show the
showninTable 5.1.The code,“Soapy”,isavailable publicly
covariance of up-link turbulence with down-link turbulence re-
and is free for use1. sultingincharacteristic“stripes”.Thetop-rightimageshowsco-
The phase screens used in these simulation have been variance between up-link turbulence, which is constant for each
madesignificantlylargerthanthesizeofthesimulatedtele- x-ycombination.
scope aperture. This mitigates the periodic nature of some
phase screen generation algorithms and ensures that there
issufficientpowerinlowordermodesatthespatialscaleof our qualitative predictions. The first term, (cid:104)sαtsβt(cid:105), (top-
the telescope aperture (?). left) is identical to that caused by down-link turbulence
and hence looks similar to those used for conventional LA
LTAO, though with down-link tip-tilt included. The final
5.2 Simulated covariance matrices term,(cid:104)sαlsβl(cid:105),(top-right)istheresultofcovariancebetween
thepathofthetwolasersandishenceconstantforeachx-y
The creation of the tomographic LGS tip-tilt reconstruc-
pair. The middle terms, (cid:104)s s (cid:105) and (cid:104)s s (cid:105), (bottom left
αt βl αl βt
tor depends upon the covariance matrices between the var-
and right) are dominated by the strong covariance between
iousWFSs,theformofwhichwasderivedinequation(16).
up-link tip-tilt on one WFS and a small number of sub-
For the results presented in this paper we simulate the co-
aperturesontheotherWFSwhichobserveitsup-linkpath.
variance matrices by recording many open loop AO system
Hence they are seen as horizontal and vertical“stripes”of
frames on uncorrelated turbulence phase screens, until the
high slope covariance.
covariance matrices converge to the theoretical case. This
process must be repeated for every atmospheric and AO
configuration simulated.
5.3 Performance versus turbulence altitude
An example simulated covariance matrix between two
LGS WFSs with up-link included, with a single turbulence In§3and§4itwaspredictedthatthemethodofincluding
layer at an altitude of 14km, is shown in Figure 3. The co- LGS up-link tip-tilt would be more effective for turbulence
variance matrix has been deconstructed into its constituent at higher altitudes. To further investigate this hypothesis,
terms by simultaneous simulation of LGS with and with- AO correction performance versus the altitude of a single
outtheeffectsofLGSup-linktip-tiltobservinginthesame turbulence layer is simulated, with results shown in Figure
direction. The final covariance matrix used to create the 4.
tip-tilt LGS tomographic reconstructor is the sum of these Inlinewithourpredictions,theresultsshowlowperfor-
terms. manceofLGSup-linktip-tiltdeterminationwhenlowlayer
FromFigure3,itispossibletoseethatthetermsmatch turbulenceispresent.Morepromisingly,theyalsoshowthat
the methods works well to correct high layer turbulence,
where the correction may even approach that of LTAO us-
1 https://github.com/andrewpaulreeves/soapy inganon-axisNGSfortip-tiltcorrection.Discontinuitiesin
MNRAS000,1–9(2015)
Tomographic laser guide star tip-tilt determination 7
1.0 16
On-axistip-tiltNGS
Uplinktip-tiltretrieval 14
Notip-tiltcorrection
0.8
12
10
0.6
o m)
Rati e(k 8
Strehl0.4 Altitud
6
4
0.2
2
0.0 0
0 2 4 6 8 10 12 14 0.0 0.1 0.2 0.3 0.4 0.5 0.6
Altitudeofturbulentlayer(km) RelativeTurbulenceStrength
Figure 4.TheStrehlratiooftomographicLGSAOwithaNGS Figure 5.Theatmosphericturbulenceprofileusedinthesimu-
tip-tilt sensor (dashed), the tip-tilt retrieval method (solid) and lations.Itcontainshighlayersofsignificantstrength,whichwill
notip-tiltcorrection(dotted)versusthealtitudeofasingleturbu- requireatomographicsystemtocorrect.Theprofilealsoincludes
lence layer. The LGS up-link tip-tilt determination performance astronggroundlayer,whichwedonotexpecttobewellcorrected
increasessharplyasthelayeraltitudeincreases. bytheLGStip-tiltdetermination.
the curve are due to the up-link beam overlapping different
numbers of sub-apertures at different altitudes. 1.0
On-axistip-tiltNGS
Uplinktip-tiltretrieval
5.4 Performance with multi-layer turbulence Notip-tiltcorrection
0.8
profile
To give an impression of how the algorithm may perform
with no NGS under more realistic atmospheric turbulence 0.6
o
conditions, we perform simulations with the profile shown ati
R
in Figure 5. This contains a strong ground layer, which is hl
e
unlikely to be well corrected by the up-link tip-tilt deter- Str0.4
mination, as well as significant higher layers, for which the
benefit of including the tip-tilt correction may become ap-
parent.
Figure 6 shows the performance of LTAO with an on- 0.2
axis NGS tip-tilt, LTAO with tip-tilt retrieval and LTAO
withnotip-tiltcorrectionversusincreasingseeingstrength.
Withastronggroundlayerpresentthemethodisclearlynot
0.0
as effective as when there is only high layer turbulence. It 0.08 0.10 0.12 0.14 0.16 0.18 0.20
does still provide a small improvement over correction with IntegratingSeeingStrength,r0(cm)
no tip-tilt correction.
Tofurtherinvestigatehowthissmallimprovementcould
Figure 6. Performance of LTAO with an on-axis tip-tilt NGS
aid spectrographic instruments, the Ensquared Energy into
(dashed),withnoNGSandthetip-tiltretrieval(solid)andwith
apotentialspectrographspaxelisplottedagainstthesizeof
notip-tiltcorrection(dotted).Thetip-tiltretrievalmethodper-
thespaxelinFigure7.FortheseresultstheFriedparameter, forms slightly better than with no tip-tilt correction, but the
r0, is 14 cm. As previously discussed, ? have explored how presence of ground layer turbulence has degraded performance
LGS AO with no tip-tilt correction may be useful for some significantly.
science cases due to the increased throughput. Our method
can provide higher throughput, still without the use of any
NGSs. For instance, a spectrograph with spaxel size of 100
mas will receive around 5% more light per spaxel for our
simulate conditions.
MNRAS000,1–9(2015)
8 A. P. Reeves, T.J.Morris, R. M. Myers, N. A. Bharmal and J. Osborn
6.2 Combination with ground layer NGS tip-tilt
correction
1.0
On-axistip-tiltNGS The greatest potential impact of this method of accounting
Uplinktip-tiltretrieval for LGS up-link turbulence is in combination with a num-
Notip-tiltcorrection ber of far off-axis tip-tilt NGS. With the LGS tip-tilt re-
0.8
trieval, high layer turbulence can be corrected for, and the
y LGScanstillcorrectforhighordergroundlayerturbulence.
g
ner This leaves only turbulence near the ground for which the
E
d0.6 AO system requires tip-tilt information. The ground layer
e
uar is common to all directions, so can consequently been cor-
q
ns rectedbyanumberofveryfaroff-axisNGS.Thiscangreatly
E
nof0.4 increase the sky-coverage of LTAO systems.
ctio Ontheotherhand,performancecouldbeimprovedfor
Fra existing LTAO configurations when using the current fur-
thestoff-axistip-tiltNGS.Inthiscasemuchofthewavefront
0.2
errorisfromhighaltitudetip-tiltmodeswhicharenotwell
sampled by the far off-axis tip-tilt NGS. If the LGS tip-tilt
up-link determination is implemented, performance may be
0.0 significantly increased.
40 60 80 100 120 140 160 180 200
In future work we will continue to further investigate
ApertureDiameter(mas)
the implementation of an AO configuration where ground
layer tip-tilt is corrected using far off-axis NGS tip-tilt ref-
Figure 7. Ensquared energy versus aperture size of LTAO with erences.Wewillalsoexaminethepotentialperformanceand
an on-axis tip-tilt NGS(dashed), with no NGS and the tip-tilt skycoveragegainsofexistingLTAOwithfaroff-axistip-tilt.
retrieval (solid) and with no tip-tilt correction(dotted). Though
never approaching that of LTAO with an on-axis NGS, the tip-
tilt retrieval improves throughput over the case when no tip-tilt
7 CONCLUSIONS
correctionisperformed.
We have demonstrated theoretically and in simulation the
viability of a tomographic LGS reconstructor which deter-
minestip-tiltinformationbyaccountingfortheup-linkpath
throughatmosphericturbulenceofeachLGS.Thisisproven
possible geometrically and implemented using a LA based
6 DISCUSSION techniquetoutiliseinformationbasedoncorrelationsofad-
jacent sub-apertures.
6.1 Improvement of LGS only AO The algorithm shows good performance when correct-
ing for high layer turbulence, close to that of LTAO with
Themethodpresentedinthispaperhasbeenshowntocor- anon-axistip-tiltNGS.Theperformancewhenlowaltitude
rectwellforhighlayerturbulenceaboveatomographicLGS turbulence is present is much degraded, though still an im-
AO system. As was predicted in the derivation of the algo- provement over having no tip-tilt correction.
rithm, it does not correct well for low layer turbulence. At Though the method may be of some use as an LGS
most major observing sites the ground layer of turbulence only AO reconstructor for low spatial order science cases,
is a significant contributor to overall seeing strength (???), wemainlyenvisageitaugmentingLTAOandMOAObyal-
especially when combined with other effects such as seeing lowing further off-axis tip-tilt NGSs and hence improving
causedbythetelescopestructureitself(??).Itisalsopossi- AO corrected sky coverage. It also allows for greater LTAO
blethataturbulenceprofilecontainingmorethanfivelayers performance with existing tip-tilt NGS seperations. In fu-
mayfurtherreduceperformancefurther.Aprofilewithonly tureworkswewillexpandonthesethemesandquantifythe
fivelayerswaschosenduetothecomputationalloadofper- available improvements.
forming physical light propagation between each layer. It is One major conclusion from this work is that LGS up-
intended to optimise the code to allow simulations with a link turbulence does not have to be simply ignored and
higher number of turbulence layers to be performed for fu- discarded as is currently the case. It is not irretrievably
ture works. With this in mind, the up-link tip-tilt method entangled with the down-link turbulence, and as such can
aloneisunlikelytoprovideAOcorrectionapproachingthat provide useful information about the atmosphere above the
of LTAO with NGS tip-tilt guide stars. telescope.
However,forsomesciencecaseswhichrequirelow-order
correctioninasectionoftheskysparselypopulatedbysuit-
abletip-tiltNGS,accountingforup-linkturbulencemaystill
ACKNOWLEDGMENTS
beofsomeuse.Wehaveshownthatitmayprovideamodest
improvement in system throughput. It can be implemented APR gratefully acknowledges support from the Science
without great hardware modification, and only requires a and Technology Facilities Council (STFC) in the form of
change in tomographic reconstructor in the real-time con- a Student Enhancement Program (STEP) award (grant
troller. code: ST/J501013/1). TJM, RMM, NAB and JO also
MNRAS000,1–9(2015)
Tomographic laser guide star tip-tilt determination 9
so
Hθ
l
Hθ −θ (H−h)
s =θ − l d
αl l H
x Hθ -x
α   H−h
=θ . (A4)
d H
APPENDIX B: CALCULATING RESOLVED
TOMOGRAPHIC BIN HEIGHTS
Only a finite number of LGS up-link offsets at a turbulent
layer can be predicted by the above method. This is a re-
sultoffiniteresolutionoftheWFSbeingused.Foracentre
launched tomographic LGS, only half the number of WFS
θ sub-aperturesviewthepathofanotherLGS,henceonlyhalf
d
can be used to predict up-link LGS motion.
Turbulence which affects the path of an LGS can only
bemeasuredinvertical‘bins’wherethemeasurementisthe
sum of the turbulence in the sub-aperture FOV within the
H
vertical bin. The heights of these ‘bins’ can be calculated
byconsideringthegeometryofthesystem,againillustrated
in Figure 2. θ , θ are the launch angles for LGS α and β
α β
respectively,H istheheightoftheLGSconstellation,D is
s
the displacement for the centre of sub-aperture n and the
S
αl LGSlaunchpositionwithisassumedtobethecentreofthe
θ
x telescope pupil, D is the diameter of the telescope pupil.
h θα For small angles
D
h= s . (B1)
θ +θ
s α
θ can be obtained by considering the displacement on the
s
ground between LGS position and the centre of the sub-
aperture, D +Hθ .
s β
D +Hθ
θ = s β. (B2)
s H
Figure A1.ThedisplacementonLGSαcausedbyaturbulence Ds is dependent on the sub-aperture of interest, n, where
encounteredonup-link. the position of the centre of a sub-aperture is (n+0.5)d,
and d is the diameter of sub-aperture.
D
acknowledge financial support from STFC (grant code: D = −(n+0.5)d (B3)
s 2
ST/L00075X/1).
Thedatausedfortheresultsinthispublicationisavail- Finally,anexpressionfortheheightofthecentreofthe
able on request to the author. resolved height bin is, h, can be obtained.
D
h= s (B4)
Ds+Hθβ +θ
APPENDIX A: FORMULATING THE H α
HD
RELATIONSHIP BETWEEN A SUB-APERTURE h= s (B5)
D +H(θ +θ )
MEASUREMENT AND UP-LINK LGS JITTER. s α β
H(D −(n+0.5)d)
To find the relationship between a sub-aperture measure- hn = D −(n+20.5)d+H(θ +θ ) (B6)
ment on WFS β, s and the measured up-link on LGS α, 2 α β
βt
s weconsiderFigureA1,whichshowstheeffectofasmall
αl
tilt on LGS α up-link.
ThispaperhasbeentypesetfromaTEX/LATEXfilepreparedby
theauthor.
s =θ −θ , (A1)
αl α x
where θ is the launch angle of LGS α and θ is dependent
α x
on the tip-tilt LGS α passes through at height h.
Hθ −x
θ = l (A2)
x H
and
x=θ (H−h), (A3)
d
MNRAS000,1–9(2015)