European Organisation for Astronomical Research in the Southern Hemisphere (ESO), Karl Schwarzschildstr. 2,D-85748 Garching b. Muenchen, Germany ([email protected])
Received August 6, 2008; revised October 24, 2008; accepted October 29, 2008;posted October 30, 2008 (Doc. ID 99882); published December 10, 2008
. INTRODUCTIONn the 55 years since Horace Babcock published his semi-al work “The possibility of compensating astronomicaleeing” [1], adaptive optics (AO) has been thought of ex-lusively as a tool dealing with atmospheric turbulence.et, before the concept of AO was created, a method foreasuring and compensating a distorted wavefront was
nitiated by opticians’ need to reduce optical aberrationsroduced by thermal and gravitational forces. Afterward,he correction of quasi-static aberrations in optical sys-ems received the name of AO so as to clearly distinguishetween the bandwidths of the two systems [2]. Moreover,f active control of the optics remained as part of the tele-cope, AO was pushed to the level of the instrumentation3], and the two conceptually equal systems achieved dif-erent statuses.
Now, in the epoch of the extremely large telescopesELTs), the two systems are merging together again in theomplex concept of an active-adaptive telescope [4]. De-igning the European Extremely Large Telescope (EELT),he specialists realized that active optics and AO can noonger be considered as two independent modules. In thective-adaptive telescope the deformable (adaptive) mir-or (DM) and the meniscus active mirror (or mirrors) arembedded into the main telescope optics. The functional-ties of active optics and AO are mixed within the inte-rated optical system of the telescope. A typical examples as follows: the wavefront measured by an AO sensor iseflected by an adaptive DM, while the DM corrects theberrations of the telescope. The strategy for global con-rol of such a system is currently being created within theramework of the EELT project [5].
One of the aspects of this complex system is a subject ofhe present paper. Our objective is to quantify the abilityf AO to compensate for the part of the distorted wave-ront produced by segmentation of the primary mirror. We
ntroduce a “segmentation error” term, under which wenderstand a wavefront composed of multiple disconti-uities. The discontinuities are due to the aberrations ofhe individual segments. The aberrations might be re-iduals of the phasing system (piston, tip, and tilt), theegment shape control with the warping harnesses,nd/or the residuals of segment polishing. The errors be-ween the different segments are assumed to be uncorre-ated, which represents a worst-case scenario for the im-ging. The AO of either the telescope or in the instrumentor both) sees these distortions and tries to compensateor them by adjusting the shape of the DM. Naturally, theore degrees of freedom AO has, the better the possible
orrection.The level of correction is evaluated through the root-ean-square (RMS) improvement criterion: the ratio be-
ween the RMS values of corrected and uncorrected wave-ronts. As the RMS is related to the power spectrum of thehase (PSD), we considered it necessary to develop a fullathematical base for the calculation of the PSD from
egmentation error and to study it in detail. That is theeason why AO appears only in the third section of theresent paper, while the whole second section addresseshe image properties before adaptive correction. In thisection will be found the general expression for the point-pread function (PSF), a discussion about its relation tohe PSD, and the functions describing the PSD for a par-icular aberration. These functions form the analyticalase of the rest of the paper. In previous publications [6,7]e spoke of the halo formed by random piston, tip, and
ilt of the segments. We now generalize it for an arbitraryegment aberration and calculate it for the first 16ernike modes. The material of this section has indepen-ent value and can be used for other tasks related to im-ge formation in the segmented pupils.Section 3 introduces the AO in a very abstract way:
odeled as a hard-edge filter of the phase. The ratio be-ween the integrals of the initial and filtered spectra givescorrection factor—the improvement in RMS. The size of
he filter defined by the cutoff frequency of AO is a freearameter in this study. Therefore, systems of differentrders—from the low-order AO to the extreme AOXAO)—are represented at the same level of abstraction.n Section 4 an argument can be found proving that theard-edge filter correction level can never be achieved inractice. We claim that the realistic RMS improvementill always be lower than the one from the hard-edge fil-
er model. Putting together the results from the differentO models, including least-square fitting, we estimate
he expected correction in terms of RMS in optimistic andessimistic cases, introducing definitions for “good” andmedium” quality AO, respectively.
In Section 5 we use an example of a high-contrast im-ging task (HCI) to demonstrate our results being usedor a particular application. The HCI, or the direct imag-ng of exosolar planets, sets extremely tough require-
ents for the quality of the telescope optics, and in par-icular for the segments. The XAO, as a part of the HCInstrument, partly corrects for segmentation error andelps to relax the requirements on the segments’ fabrica-ion and control. Using the definition of the best RMS ob-ained with the hard-edge filter model and the assump-ion of “good” and “medium” quality of XAO, we calculatehe initial level of the segmentation error required to pro-ide extremely high �10−8� contrast in the final image.
. DIFFRACTION HALO BY A SEGMENTEDELESCOPE. PSF for the Segmentation Errore consider a segmented telescope with all identical seg-ents of a given shape. The phase error on each indi-
idual segment can be represented as a set of aberrations,escribed, for example, by a Zernike polynomial. We usehe term segment aberration to refer to a particular aber-ation of an individual segment and the term segmenta-ion error to refer to the shape of the wavefront reflectedy the whole segmented mirror. At first we do not specifytype of aberration; it allows us to present a model of dif-
raction in a general way. In this paper, we study the ef-ect of different aberrations separately; therefore the seg-entation error is described by only one polynomial Znith the coefficients ajn randomly distributed over thehole population of the segments:
��x� = �j=1
N
ajnZn�x − rj���x − rj�. �1�
n this expression x is a coordinate in the pupil plane, rjs the center of a segment with an index j, Zn is a polyno-
ial with an index n, and N is the total number of seg-ents. The function ��x−rj� is a segment transmission
unction equal to unity within the boundary of a segmentnd zero outside it.The complex amplitude in the pupil plane is described
y a similar expression. Due to the specific properties ofhe segment transmission function, the latter can be sets a multiplier of an exponent:
F�x� = �j=1
N
��x − rj�exp�ajnZn�x − rj��. �2�
he image in such a mirror is described by a PSF, whichs a modulus square of the Fourier transform of Eq. (2)8]. After a variable exchange �=x−rj and some algebraic
anipulation, the expression for the PSF takes the form
PSF�w� =1
N2A2�j=1
N
�k=1
N � d2����� � d2�������
�exp�i2�
�w�� − ����exp�iajnZn����
�exp�− iaknZn�����exp�i2�
�w�rj − rk�� . �3�
ere w is an angular distance. The PSF is normalized byhe peak intensity, �AN /��2, where A is the area of an in-ividual segment and AN is the area of the whole mirror.The next step is to introduce the ensemble averaging
or the function from Eq. (3). We assume that the valuesf the aberration coefficients ajn for any pair of segmentsre uncorrelated. We also assume that the statisticalroperties of the aberration coefficient do not change fromne segment to another. The functions of these randomalues are also uncorrelated, and the averaging of theerm in Eq. (3) containing these values gives
exp�iajnZn����exp�− iaklZk�����
= �exp�ian�Zn��� − Zn����� for j = k
exp�ianZn����exp�ianZn����� for j � k� .
�4�
e omitted the segment index in the averaged functionssing the assumption made above.To obtain an expression for the ensemble-averaged
SF, we substitute Eq. (4) into Eq. (3). After reorganizinghe terms and introducing three functions,
F�w� = � 1
N�j=1
N
exp�i2�
�wrj��2
,
B�w� = � 1
A � exp�ianZn��������exp�i2�
�w · ��d2��2
,
D�w� =1
A2 �� exp�ian�Zn��� − Zn�����
�exp�i2�
�w · �� − �������������d2�d2��, �5�
e can write the ensemble-averaged PSF in canonicalorm [9]:
PSF�w� = GF�w�B�w� +D�w� − B�w�
N. �6�
e do not use rectangular brackets for the PSF and theSD. Whether it is an ensemble-averaged characteristicr not should be clear from the context.
The canonical form for the PSF has two terms: the firstne describes the diffraction pattern and its modification,nd the second one describes the halo. In the more com-lex case when the segment shape is given by more thanne polynomial, the ensemble-averaged PSF preserveshe canonical form of Eq. (6). In that case, functions B�w�nd D�w� have a more complicated form, containing meanalues, variances, and correlations of the aberration coef-cients. We do not consider this case here.The physical meaning of a halo from segmentation er-
or is described in [9]. The image contains quasi-staticpeckles, each having the size of the Airy disk from theelescope. The speckled field is spread over an area in-ersely proportional to the size of a segment, i.e., an areauch larger than the Airy disk. For another realization of
egmentation error, the speckles are found in differentlaces, but the general outline of the speckled field is pre-erved. The shape of the statistically averaged speckledeld is described by the second term of Eq. (6), which, innalogy to the long-exposure PSF from atmospheric tur-ulence, we call halo. Using knowledge about speckle sta-istics, one can derive the high-order characteristics of thepeckles from the halo, such as intensity variance andkewness [10], but this subject is outside the scope of thisaper.We are interested in the shape of the halo itself. Let us
onsider the case when the mean value of the aberrationss zero and the values themselves are small an�1. Weeep only the second-order terms in the exponent expan-ion:
exp�ianZn����exp�ianZn����� � 1 − 12�n
2�Zn2��� + Zn
2�����,
exp�ian�Zn��� − Zn����� � 1 − 12�n
2�Zn2��� + Zn
2�����
+ �n2Zn���Zn����. �7�
e have introduced �n for the standard deviation of anberration value: �n
2 = an2. This characteristic is related to
he wavefront RMS, as demonstrated in Appendix A. Sub-tituting Eq. (7) into Eq. (5) after some algebraic manipu-ations, we obtain an approximate expression for the halo:
D�w� − B�w�
N�
�n2
N�tn�w��2, �8�
here
tn�w� =1
A � Zn�������exp�i2�
�w��d2�. �9�
he halo formed by the segmentation error in a second-rder approximation is proportional to the variance of theegment aberration coefficient, inversely proportional tohe number of segments, and its shape is a modulusquare of the Fourier transform of the polynomial corre-ponding to this aberration within a segment’s boundary.ote that to obtain Eq. (8) we did not use any particularistribution law of the aberration coefficients.
. PSD for the Segmentation Errorhe PSD or energy spectral density of the phase error is aodulus square of the Fourier transform of the phase:
PSD�w� = � 1
AN � ��x�exp�i2�
�w · x�d2x�2
. �10�
he normalization is the same as for the PSF. Substitut-ng ��x� from Eq. (1) and using variable exchange �=xrj, the expression for the PSD becomes
PSD�w� =1
N2�j=1
N
�k=1
N
ajnakn exp�i2�
�w�rj − rk��
� � 1
A � Zn�������exp�2�
���d2��2
. �11�
n the last term we recognize the modulus square of theunction tn�w� from Eq. (9). Because ajna=0 for j�k,a=0, and an
2=�n2, the average of the sum is equal to
n2 /N. The ensemble-averaged PSD coincides with theight-hand side of Eq. (8):
PSD�w� =�n
2
N�tn�w��2. �12�
herefore, the second-order term of the PSF halo is, as ex-ected, the statistically averaged PSD of the phase error.
. PSD for a Zernike Aberration in a Hexagonalegmentrom now on we use the term “halo” to refer to theecond-order approximation of the smooth part of thensemble-averaged PSF, which coincides as we have seenith the phase PSD. We have established that the halo is
inearly proportional to the variance of the aberration co-fficient and inversely proportional to the number of seg-ents. In this section, we calculate the shape of the halo.The segments considered from now on are regular
exagons. The flat-to-flat width of the hexagon we call d.berrations are the Zernike polynomials orthogonal in aircumscribing circle of the hexagon. Cartesian coordi-ates are chosen so that the axis x is aligned with theorner-to-corner direction of a hexagon. Note that this co-rdinate system is rotated by 90° with respect to the onesed in our referenced Thirty Meter Telescope (TMT) pa-ers [11–13]. When we later compare our results to theirs,e take this fact into account.We introduce scaled coordinates in the image plane
ith a simultaneous change of the coordinates in the pu-il plane:
= �dwx/2�, = �dwy/2�,
x = �3�x/d, y = �3�y/d, �13�
here � is the wavelength, wx and wy are projections ofhe vector w, and �x and �y are projections of the vector �.he functions tn in the new dimensionless coordinates are
tn�,� =2
3�3�
−�3/2
�3/2
dy�−1+�y�/�3
1−�y�/�3
dxZn�x,y�
�exp�i�x + y�4/�3�. �14�
he polynomials Zn�x ,y� are the classical Zernike polyno-ials orthonomal in a circle of unit radius. Integrals of
his type can be calculated exactly, and analytical expres-ions for the functions tn� ,� exist. They possess very in-eresting symmetrical properties [10], but it is beyond thecope of our paper. Figures 1 and 2 show the functionstn� ,��2 for the first 16 polynomials, whose expressionsn traditional polar coordinates are given in Table 1.
. ULTIMATE PERFORMANCE OFDAPTIVE OPTICS. Relation between PSD and RMS
f we assume that the wavefront has a zero mean, the ex-ression for the RMS is
Fig. 1. Normalized PSD of segmentatio
RMS2 =1
AN � �2�x�d2x. �15�
pplying Parseval’s theorem on the Fourier transform re-ating the PSD and the phase [Eq. (10)], we obtain the fol-owing expression for the RMS:
RMS2 =AN
�2 � PSD�w�d2w. �16�
O modifies the PSD. The modification of the PSD de-ends on the particular design of the AO system. The areahere the PSD is modified is called the AO control regionnd is limited by the DM cutoff frequency. We call this re-
ion GAO. Outside the control region, i.e., in a complementf GAO in R2, the PSD remains unaltered.
The RMS values before and after correction are
MSini2 =
AN
�2 �GAO
PSDini�w�d2w +AN
�2 �GAO
cPSDini�w�d2w,
�17�
MSfin2 =
AN
�2 �GAO
PSDfin�w�d2w +AN
�2 �GAO
cPSDini�w�d2w,
�18�
here PSDini and PSDfin are the PSDs before and afterorrection, respectively. We used the fact that outside the
Fig. 2. Same as in Fig. 1 but for aberration
ontrol region, i.e., in GAOc , the PSD does not change.
. Best Correction Factorn ideal AO system completely cancels the PSD insideAO. This type of AO is usually simulated as a top-head
patial filter applied on the initial phase [13,14]. The finalMS in this case simplifies to
RMSfin,ult2 =
AN
�2 �GAO
cPSDini�w�d2w. �19�
ustituting Eq. (12) into Eq. (19), we obtain the final RMSiven the type of segment aberration, the variance of theberration coefficient, and the number of segments.
From Eqs. (12), (17), and (19) we conclude that a fun-amental parameter of this model of AO is a ratio be-ween the final and the initial RMS values. According tohe adopted notation [15], we call this ratio a correctionactor, �AO,n. The lower the correction factor, the betterhe performance of the AO. The improvement factor cane estimated as 1−�AO,n.The case of the hard-edge filter is characterized by the
est correction factor:
�AO,nult =
RMSfin,ult
RMSini. �20�
his value is independent of the initial RMS, the numberf segments, and the wavelength. In the sense of a bestase, any realistic AO delivers a correction factor thatoes not exceed the best correction factor defined above:AO,n��AO,n
ult .To calculate the best correction factor, one needs to
now only two inputs: the type of aberration and the ge-metry of the control region. Suppose that the control re-ion is rectangular and bounded by a cutoff frequency ofhe DM, ±� /2da, where da is the distance between actua-ors projected onto the pupil plane. In the normalized co-rdinates �� the cutoff frequency is �� /4��d /da�. Thus,he main parameter is the ratio between the segment sizend the interactuator separation.Substituting Eq. (12) into Eqs. (17) and (19) after the
oordinate normalization, we obtain the following expres-ion for the best correction factor in the case of hexagonalegments and a rectangular control region:
�AO,nult = �1 −
2�3
�2 n�
−�d/4da
�d/4da �−�d/4da
�d/4da
�tn�,��2dd�1/2
.
�21�
Table 1. Zernike Polynomials
Aberration Name Expression
iston 1ip 2� cos���ilt 2� sin���efocus �3�2�2−1�stigmatism X �6�2 cos�2��stigmatism Y �6�2 sin�2��oma X �8�3�3−2�2�cos���oma Y �8�3�3−2�2�sin���pherical �5�6�4−6�2+1�refoil X �8�3 cos�3��refoil Y �8�3 sin�3��econd astigmatism X �10�4�4−3�2�cos�2��econd astigmatism Y �10�4�4−3�2�sin�2��econd coma X �12�10�5−12�3+3��cos���econd coma Y �12�10�5−12�3+3��sin���econd spherical �7�20�6−30�4+12�2−1�
unctions tn� ,� were calculated in Subsection 2.C. Pa-ameter n is a form factor relating RMS to the variancef the aberration value. It is a rational number presentedn Appendix A.
The best correction factor for the first 16 Zernike poly-omials is shown in Figs. 3–5 as a function of the actua-ors’ density (thick curve). As one can see, the aberrationrder in terms of RMS improvement is not the same forhe low-order AO �d /da�2� and the extreme AO �d /da6�. Aberrations can be split into groups by the order ofMS improvement. Piston is the best correction aberra-
ion for any regime. For low-order AO the groups are pis-on (60% improvement); tip and tilt (40% improvement);efocus (30% improvement); astigmatism Y, coma X, andoma Y (20% improvement); astigmatism X and secondstigmatism Y (12% improvement); and spherical aberra-ion, second astigmatism X, trefoil X, trefoil Y, sectionoma X, section coma Y, and second spherical aberrationre almost uncorrected (1%–5%).In the case of extreme AO, the order changes. We can
onventionally select the groups of aberrations: piston,ection astigmatism Y and spherical aberration (80% im-rovement); coma X and coma Y (76% improvement); sec-nd astigmatism X, second coma X, section coma Y, andefocus (74% improvement); tip and tilt (72% improve-ent); astigmatism X, astigmatism Y, and second spheri-
al aberration (64% improvement); and trefoil X and tre-oil Y (55% improvement).
The discrepancy between X and Y components of theame aberration are explained by the fact that for theexagonal geometry, the X and Y directions are notquivalent: rotation of a segment by 90° does not transferne component of the aberration into another.
. REALISTIC PERFORMANCE OFDAPTIVE OPTICS. Discussioneal AO includes many sources of error [16]: DM fitting,patial aliasing, servo lag, photon noise, detector noise,nd anisoplanatic error. Due to this, the residual RMS isigher than the one in our best case. For us it means thathe PSF of the residual phase will have a remainingecond-order halo inside the control region. For the seg-entation error, temporal errors are negligible: there iso delay between the measurement and the correction forhe quasi-static wavefront. For the same reason the pho-on noise is not an issue: long exposures can be used toncrease the signal-to-noise ratio. The problem ofnisoplanatism does not arise here because the physicalource of the wavefront distortions is located in one planenly. Therefore the main source of residual AO error forhe segment aberrations is DM fitting. The fitting errorrises when a continuous DM with given properties (ac-uator spacing and geometry, influence function shape,oupling, etc.) reconstructs a wavefront.
There exist three principal models of DM action on theavefront: modal control, when the low-order aberrationsf the whole mirror are completely corrected; filtering ap-roach, when the DM influence is described as a low-pass-lter on the phase [17]; and zonal model, when the DM is
ocally deformed by displacements of the actuators.
The modal control neglects the effect of the pupil bor-er. It is applicable to the monolithic mirror case, wherehe border is outside the imaging pupil. It is not appli-able to the case of segmentation, where the pupil partsontaining the segment borders play a role in image con-truction and hence affect the image quality. The modalpproach therefore does not fit our task.The filtering approach is barely applicable here either.
he spectral corrective ability of AO cannot be describedn terms of a transfer function. The actuators exist in spe-ific locations, and the mirror does not respond in a shift-nvariant fashion. This is shown in [18]. The argument iss follows: imagine a wavefront shifted laterally in the op-ical plane. The shifting has no effect on the spectrum of
ig. 3. Correction factor: comparison among the best case (thicktriangles and stars). The dashed and dotted–dashed curves are tiston to astigmatism Y.
he incident wavefront. However, the least-square fit ofhe mirror surface will not be the same for the shifted andnshifted wavefronts due to the inability of the mirror toatch the shift. The reflected spectrum is different in
ach case, so the transfer function is not unique.If one can use the filtering approach as an approxima-
ion for a homogeneous wavefront, such as a turbulent at-osphere, for an inhomogeneous wavefront (such as seg-entation error), the situation is even more critical. In
his case not only is the DM not invariant by the transla-ion, but the wavefront is not as well. The lateral shifthanges the geometrical properties of the wavefront, un-ess this shift is equal to an integer number of segmentizes.
curve), a least-square fit (circles), and the results of simulationscases degraded by 10% and 20%, respectively. Aberrations: from
Nevertheless, we will now assume that a realistic AOystem preserves the linear property of the ideal one: as-ume that the correction gain does not depend on the ini-ial value of the error and that the performance can beuantified by the correction factor. What is the value ofhe correction factor for a realistic AO and how it is re-ated to a best case?
There are few studies dealing with the AO correction onegment aberrations with available data. The most valu-ble for our study is a paper by Crossfield and Troy [13]. Itives correction factors for two models of AO: end-to-endimulated wave optics and the hard-edge filter. The pre-iously mentioned linearity of AO in a range of small er-ors is also demonstrated there. Our previous work [15]ives the results for the DM-fitting AO model for piston
Fig. 4. Same as in Fig. 3 but for aberratio
egment aberration only and without using the least-quare model. We now further this method by includingeast-square fitting and more aberrations.
. Least-Square Fithe DM response is composed of influence functions withredefined profiles. We assume that the influence func-ion for each actuator is the same. To avoid ambiguity, wellow no actuators to be placed between the segments. Toerive the general expressions independent of a particu-ar mirror configuration, the fitting procedure is appliedo each segment separately. This approach allows us tovoid performing simulations of the segmentation error
nd to use a quasi-analytical approach. The drawback ishat in this method the fitting of phase steps is not opti-ized.Let IF�x−xm� be an influence function centered at
oint xm in the system of coordinates bounded with theenter of the mirror. As usual, we consider the aberrationseparately, so the residual wavefront after the correctionecomes
�res�x� = �j=1
N
ajn�Zn�x − rj���x − rj� − �m��j
�mnIF�x − xm�� .
�22�
he notation m��j means that the sum is taken only overhe actuators located within the area of segment j. The co-fficients �mn are the actuator response to an aberrationf unit value: we assume that a correction is linear formall values of ajn. The values of the coefficients �mn de-end on the aberration index and the position of the ac-uator.
The RMS of the residual wavefront is given by Eq. (15)ut with the phase from Eq. (22). Changing the variables�=x−rj, �m=xm−rj) and performing averaging, for the fi-al RMS we obtain
RMSfin2 =
�n2
N �j=1
N 1
A � �Zn������� − IFsjn����2d2�. �23�
Fig. 5. Same as in Fig. 3 but for aberrations f
ollowing [15] we have introduced the segment influenceunction
IFsjn��� = �m��j
�mnIF�� − �m�. �24�
he segment influence function is the DM response whenhe wavefront is distorted by the nth aberration of unitalue introduced on one segment only. As the actuator’sonfiguration, and hence the set of coefficients �mn, mighthange from one segment to another, IFsj��� is a paramet-ic function of the segment index. Normalizing Eq. (23) byhe square of the initial RMS, for the correction factor webtain
�AO,n =�RMSfin2
RMSini2 =�RMSfin
2
n�n2
=� 1
N�j=1
N 1
A n� �Zn������� − IFsjn����2d2��1/2
.
�25�
o minimize �AO,n one must find for each configuration ofctuators a set of coefficients �mn minimizing the integral
et us sample a wavefront in the pupil plane �=�i and in-roduce a matrix Him=IF��i−�m� and a vector BiZn��i����i�. Then the vector of the coefficients �mn�m1,2, . . . � minimizing a norm of the vector H�−B is theell-known least-square solution [18,19]
� = �HtH�−1HtB. �27�
he symbol t is the matrix transpose. The sampling areaust exceed the area of the segment by a minimum of 2da
rom every side. When the number of sampling points in-reases, the solution asymptotically approaches the exactolution of Eq. (26).
The residual error is given by the following expression: i
TfrtcmFtugtdf
FwE
ej2 = B�I − H�HtH�−1Ht�Bt, �28�
here I is the unitary matrix. We keep index j to reminds that the residual error depends on actuator position.fter substituting this solution into Eq. (25) for the cor-
ection factor, we obtain
�AO,nfit =� 1
N�j=1
n ej2
nA, �29�
here A is a number of sampling points covering the seg-ent transmission function.We have performed the calculation of �AO,n
fit for hexago-al segments, Zernike polynomials, and the cubic spline
nfluence function. The one-dimensional projection of this
nfluence function is given by [20]
IF�x� =1
�2c + 1��1 + �4c − 2.5�x2 + �− 3c + 1.5��x�3 0 � x � da
�2c − 0.5��2 − �x��2 + �− c + 0.5��2 − �x��3 da � x � 2da,
0 x � 2da� �30�
here c�Re is an interactuator coupling coefficientshape parameter). In our algorithm we used 25% cou-ling �c=0.25�.According to Eq. (29), the correction factor must be av-
raged over all existing actuators distributed over a hexa-on. We use four extreme configurations. In the first con-guration the central actuator is placed in the center of aexagon, and the actuator grid is collinear with the sys-em of coordinates used to define the aberrations. The sec-nd configuration is obtained from the first one by a shiftf a half-period in actuator spacing in the Y direction. Thehird is obtained from the first one by a shift of a half-eriod in the X direction. The fourth is obtained in theame way by shifting a half-period both in the X and Yirections. For the sampling area we used a grid with00�300 points on a square 1.7d�1.7d; A=25,959. Theesults are presented in Figs. 3–5 (circles).
As we already mentioned, the algorithm is not optimalhen each segment is fitted separately. We also per-
ormed simulations fitting the whole phase map repre-enting a particular realization of the segmentation error.o distinguish it from the method above, we call it globaltting. To simulate the segmentation error, we use theodel of the EELT primary mirror [5] with 984 segments
nd populate the aberration randomly over the segmentscentral normal distribution). The size of each segment is.22 m. The model of the DM corresponds to the DM em-edded into the EELT optics with an actuator separationf 50 cm (projected onto the primary mirror): d /da=2.44.he one-dimensional projection of the influence function
s [21]
IF�x� = �1 − �x�3.805 + 3.74 ln��x���x�2.451 0 � x � da.
0 x � da��31�
he initial and residual phase maps for the piston and de-ocus aberrations are shown in Fig. 6. Note that the erroremains near the segments’ borders—at the locations ofhe wavefront discontinuities. The correction factor is in-luded in Figs. 3–5 (triangles). This algorithm yieldsuch better results than fitting an individual segment.or the low-order AO, a separate segment-to-segment fit-
ing gives an unrealistic estimation and should not besed. Although we do not have at our disposal results forlobal fitting with a higher density of actuators, we expecthat for the extreme AO case two ways of fitting will pro-uce similar results. We plan to return to this issue in theuture.
ig. 6. Initial and residual phase maps from the simulationith global least-square fitting. Images show one-quarter of theELT primary mirror.
. “Good” and “medium” Quality Adaptive Opticshe correction factor obtained from different models isollected in Figs. 3–5. On each plot the thick curve is theest correction factor, where circles represent the one ob-ained from least-square fitting when the aberration onach segment is fitted separately. Triangles are the re-ults of the global fitting simulations. Stars are the re-ults of the TMT simulation taken from the work byrossfield and Troy [13]; the upper star is the result of theave-optics simulation using the Arroyo library. Theysed a Shack–Hartmann model with a 127�127 actuatorM and a spatial-filtered Shack–Hartmann wavefront
ensor acting on the unobscured TMT aperture �d /da4.4�. The lower star is the result from the same paperbtained with a hard-edge high-pass filter. By definition,hese results must coincide with our best correction fac-or. Some discrepancy is explained by numerical errorhile simulating the high-pass filter or by inexact readingf the plot in the paper (exact numbers were unavailableo us). We took care on the swapping of the X and Y com-onents for the aberrations when placing the TMT resultsn our plots. To the aberrations calculated by Cossfieldnd Troy, we also added the second spherical aberrationnd second coma, because we believe that aberrationsith a lower azimuth order will dominate over aberra-
ions with a higher azimuth order. That is confirmed byhe test with the Southern African Large TelescopeSALT) segments [22].
One of the main applications of this study is establish-ng the requirements of the segmentation error for anLT. As we cannot predict the AO technology of the fu-
ure, for technology- and model-free estimation we intro-uce the concept of “good” and “medium” quality AO. Thegood” quality AO addresses the optimistic assumption of10% loss in performance with respect to our best case.he “medium” quality AO addresses the pessimistic as-umption of a 20% performance loss. The correspondingMS improvement is
1 − �AO,ngood = 0.9�1 − �AO,n
ult �,
1 − �AO,nmedium = 0.8�1 − �AO,n
ult �. �32�
n the graphs in Figs. 3–5, we plot curves correspondingo the correction factor for the two cases. Notice that forhe extreme AO case the results of a least-squareegment-by-segment fitting fall into the “optimistic” cat-gory. The results of the wave-optics simulation are either0% or 20%, depending on the type of aberration as wells on the results of the global fitting. As segments-by-egment fitting, global fitting, and TMT simulations wereerformed with completely independent AO modes (andy different people), we are confident that the 10%–20%ange covers most cases.
. MEAN LEVEL OF THE RESIDUAL HALOhe difference between the correction factor of the realis-ic AO and our best correction factor defines the integralalue of the residual PSD within the control region. Fromhe relation connecting the PSD and RMS, we can express
he mean level of the halo inside the control area throughhe initial RMS and the correcting factor:
PSDfin = ��AO,n2 − ��AO,n
ult �2�RMSini
2
N
2
�3
1
�d/da�2 . �33�
he initial RMS is expressed in radians. Earlier we de-ned “good” and “medium” AO by setting a certain levelor the correction factor. From Eq. (33) the mean level ofhe residual PSD for the “good” and “medium” quality AOecomes
PSDfingood =
2
�3�0.01 + 0.18 · �AO,n
ult − 0.19 · ��AO,nult �2�
�RMSini
2
N
1
�d/da�2 ,
PSDfinmedium =
2
�3�0.04 + 0.32 · �AO,n
ult − 0.36 · ��AO,nult �2�
�RMSini
2
N
1
�d/da�2 . �34�
nversely, one can use Eq. (34) to establish the level of thenitial RMS that provides the residual PSD with the re-uired mean level. Finally, let us demonstrate how the re-ults can be used in practice. As an example, we have cho-en the application of HCI [23]. The purpose of a HCInstrument is to directly image an exoplanet against theright background of its parent star. The challenge is inhe extremely high ratios between the intensity of thearent star and its planet. Typically, this ratio is about0−7–10−9, depending on the type of planet and the dis-ance to its system. The background has a number of ori-ins. It is from the diffraction of the image by the tele-cope pupil, noise due to the atmosphere (for ground-ased systems), photon noise, zodiacal noise, detectoroise, etc. The strong limitation is set by the static speck-
es created by imperfections in the optics of the telescopend inside the instrument itself. Several consecutive mod-les of the instrument serve to improve contrast, removehe static and dynamic errors, and detect a planet. One ofhe modules is extreme AO.
The distorted wavefront containing the star and thelanet after entering the instrument is corrected by thextreme AO. This is “extreme” AO in the sense of a highensity of actuators and a high temporal bandwidth. Thiss conventional AO in the sense that the wavefront is
easured and corrected in a pupil plane. The actuatoritch in the extreme AO case is about 20 cm. For a seg-ent size of 1.22 m, we obtain d /da=6. The number of
egments is 984.We define the requirements on the segment aberration
y fixing the level of residual halo after the extreme AOodule. This component of the wavefront error is not al-
ered by a coronagraph (if the coronagraph does not be-ong to the interferometric type). We ask the level of theesidual halo to be equal to the level of the residual dif-raction delivered by a coronagraph. This level is typically0−7 times the peak intensity of the initial nonaberrated
mage. Each aberration shall introduce no more than 10%f this level; i.e., the halo level from each separate aber-ation shall be 10−8. (This number might vary based onhe requirements from a signal extraction algorithm. Oururpose here is to demonstrate our method.) With in-reasing aberration order, the initial level of the PSD low-rs, but the correction ability lowers as well, and hencehe required level of the initial RMS for different aberra-ions is similar. Solving Eq. (34) with PSDfin=10−8, we ob-ained RMSini�20 nm for “good” quality AO and RMSini14 nm for “medium” quality AO in the H band �1.6 �m�.
. CONCLUSIONn a complex system of an active-adaptive telescope with
segmented primary mirror, the wavefront error intro-uced by the aberrations of the segments will be partlyorrected by AO embedded into the telescope optical sys-em and into the instrument. The level of this compensa-ion depends on the spatial spectrum of the phase and onhe number of degrees of freedom of the adaptive system.or the worst-case scenario of white noise, we calculatedhe ratio between the initial and the final RMS values—aorrection factor—for the first 16 Zernike polynomials.
We compared several models of AO: a hard-edge high-ass filter, a least-square fitting, and a full wave-opticsimulation. The least-square fitting was performed ana-ytically, considering each segment separately and in aimulation where the algorithm is applied to the wholeavefront. Gathering together the results of these mod-ls, we classified the performance of the AO into three cat-gories: best, good, and medium quality. The best categorys represented by a hard-edge high-pass filter, the gooduality AO loses 10% in performance with respect to theest case, and the medium one loses 20%.The main parameter of the study is the order of the AO,
xpressed as a ratio between the segment size and the in-eractuator pitch, m. For the low-order AO with m�2 andhe extreme AO m�4, the sensitivity to aberrations is dif-erent. If for the low-order AO the order of sensitivity fol-ows the order of polynomials (piston, tip, tilt, defocus,stigmatism, coma, etc.), then for the extreme AO the or-er changes. For example, for m=6 the best RMS im-rovement for the spherical aberration is better (80%)han for the defocus (74%). This tendency is preserved inhe least-square fitting model also, although with differ-nt values: for spherical aberration it is 71%, and for theefocus it is 65%.The classification of AO provides a tool to estimate the
nitial RMS required to reach a certain level of contrast.e performed an estimation with m=6 and a contrast of
0−8. Assuming good quality AO, we found that for eachberration the required RMS is an �20 nm wavefront; foredium quality, it is an �14 nm wavefront.Different AO models, although resulting in similar val-
es for the correction factor, give different spectra of theesidual phase and hence different shapes of the residualalo. The analysis of the shape of the residual halo is a
ogical continuation of the study presented in this paper.
PPENDIX A: FORM FACTORere we derive a relation between the RMS of the wave-
ront and the standard deviation of the aberration coeffi-
ient. We consider a wavefront with a zero-mean value.ombining Eqs. (1) and (15) for RMS, we can write
RMS2 =1
AN�j=1
N
anj2 � ��x − rj�Z2�x − rj�d2x
=1
A � ����Z2���d2� ·1
N�j=1
N
anj2 . �A1�
e have used the property of the segment transmissionunction and shifted to the segment coordinates. The firstultiplier does not depend on the aberration value. It is a
orm factor equal to the integral of the square of a poly-omial over a hexagonal surface. In the normalized seg-ent’s coordinates, the form factor is given by
n �1
A � ����n2Zn
2���d2� =2
3�3�
−�3/2
�3/2
dy�−1+�y�/�3
1−�y�/�3
dxZn2�x,y�.
�A2�
he statistically averaged RMS is therefore
RMS2 = n ·1
N�j=1
N
anj2 = n�n
2 . �A3�
e assumed that the mean value of the aberration coeffi-ient is zero. For a mirror with a large number of seg-ents, the sum over all segments normalized by theirumber can be approximated by the variance of the coef-cients: the average over the segment population for a
arge number of segments is approximated by the statis-ical average. The RMS for a particular realization of theavefront is
RMS2 � n�n2 . �A4�
able 2 shows the values of the form factor for the aber-ation considered thoughout the paper.
Table 2. Form Factor
Aberration Form Factor
iston 1ip 5/6ilt 5/6efocus 4/5stigmatism X 7/10stigmatism Y 7/10oma X 337/420oma Y 337/420phere 61/70refoil X 103/140refoil Y 9/20econd astigmatism X 257/315econd astigmatism Y 257/315econd coma X 208/231econd coma Y 208/231econd sphere 27,299/30,030
CKNOWLEDGMENTShe study is done within the framework of The Exo-Planemaging Camera and Spectrograph (EPIC) project for theELT. The simulations of global least-square fitting wereerformed by Miska LeLouarn. The author is grateful toerome Paufique for helpful discussions and a number ofise suggestions concerning the presentation of the ma-
erial. I also want to thank Liam McDaid for correctingy mistakes in English.
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