Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.arcetri.astro.it/~salinari/ELT/Yaitskova_et_Al_2002.pdf Äàòà èçìåíåíèÿ: Thu Aug 28 14:17:44 2003 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 04:14:00 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: optical telescope

DIFFRACTION IN OWL: EFFECTS OF SEGMENTATION AND SEGMENTS EDGE MISFIGURE
N. Yaitskova*a, K. Do hlenb, P. Dierickxa a Eu rop ean Southern Ob servatory b Labo ratoire d 'Astrop hysiqu e de Marseille

ABSTRACT
We consider the diffraction by segmented apertures with a very large nu mber of segments, con centrating mostly on the effects which lead to the appearan ce of regular patterns of diffraction peaks, specific to extremely large telescopes (ELT). These effects are associated to gaps between segments, turned down/up segment edges and rand om tip-tilt errors. We discuss briefly the effect of segment piston error, which does not produce the higher-order diffraction peaks, but also affect the image quality. We deli ver an analytical expression of the Point Spr ead Function (PSF) associated to each particular case and approximate formulae for the Strehl ratio and relative intensity of higher-ord er diffraction peaks. Keywords: segmented mirrors, ELT, OWL, diffraction, PSF.

1. INTRODUCTION
Diffraction effects caused by an aperture with a large number of segments (hereafter strongl y segmented aperture) differ qualitatively from th ose associated to apertures with a few segments. A segmented surface acts as a giant bidimensi onal diffraction grating, producin g an inter ference pattern with regular peaks.1-4 Although the intensit y of higher-order diffraction peaks is 10-4 ­ 10-5 times the intensity of the central peak, in a giant telescope higher order effects mu st be accounted for. This is particularly i mportant when looking for extra solar planets, which have a ratio of brightness to their parent star of ~10-9. Moreover, while for a 1.5m flat-to-flat hexagonal segment and a wavelength of 0.5µm the first peak occurs at 0.08 from the central peak, Earth-like plan ets are expected at 0.1 - 0.2 from their parent star.5 Therefore, knowledge of the position and relative intensity of the diffraction peaks is essential. Th e first part of the present paper is a general description of the PSF for a segmented telescope. Next, we consider the effect of gaps, turned up/down edges, and tip-tilt errors, emphasizing the diffraction effects. We discuss the piston error effect onl y briefly, because it does n ot lead to the appearance of a regular diffraction pattern and therefore is not particular for the ELT. Our final goal is to find the Strehl ratio and the relative intensity of higher-order diffraction peaks at differ ent angular distances. The results of all considered effects are presented in a table useful for practical applications.

2. PSF FOR SEGMENTED TELESCOPE
We consider (Figure 1) a segmented primary mirror consisting of N segments with a segmentation geometry similar to that of the Keck, i.e., organized in M concentric hexagonal "rings" around the central segment. The tot al number of segments (including the central one) is given by N = 3M(M+1)+1. Th e expression of the PSF for a segmented mirror is presented in several recent pu blications. form:
AN 1 PSF (w ) = z N
2

1-4

In general it takes the
2


j=1

N

2 A 1 w rj j exp i z AA

j



2 j ( ) exp i j ( ) exp i w d 2 . z

[

]

(1)

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Here w is the posit position vector, the area of the segment with the center-to-c

ion vector in the image plane, z is the focal real area, transmission and ph ase functions A, which for hexagonal segments is 3d 2 2. enter distance between contigu ous segments

distance, is the wavelength; rj, Aj, j, and j are the of the jth segment, respectively. We introduce the ideal A flat-to-flat width d of an ideal segment coincides in Y-direction.

In the case of a perfectly phased telescope without intersegment gaps and with optically perfect segments (j=0, Aj=A and j=, for all segments), Eq. 1 takes the form:

AN 1 PSF (w ) = z N

2


j=1

N

2 exp i w rj z

2

1 AN 2 ( ) exp i w d 2 = GF (w ) PSFs (w ). A z z

2

2

(2)

As in the case of a diffr action grating, the PSF of a segmented mirror can be represented as the product of two factors, regardless of the segmentation geometry: a "grid factor" (GF) which is the Fourier transform of the segmentation grid, usually a periodic fun ction of sharp peaks, and the point spread function of an individual segment (PSFs). With a hexagonal geometry the analytical expression for GF is1,3:

G F ( w ) = sin (3M + 1) + (M + 1

[

)

3

]

sin (M + 1 ) + 3 + sin (3M + 2 ) - M 3 , N sin (2 )sin + 3

[

][

sin M - 3 N sin (2 )sin - 3

[(
( (

(

)]
2

)] )

)

(3)

where

= (d 2z )w x and = (d 2z )w y are normalized coordinates in the image plane. The grid factor corresponds to a hexagonal grid of sharp peaks wh location is defined by the conditions: w y = m z d 2 = m , or ± 3 = n w y ± 3w x = n 2z d where n and m are integer values. For the following analysis it is important to point out that the value of all peaks in GF equals to unit y regardless of M. The geometry of the GF is sh own in Figure 2. (To con vert the coordinates in image plane into units of radians we must omit a focal distance in Eqs. 5, i.e. substitute z/d by /d.)

(4) ose (5) the the

Because of the shape of the hexagonal grid, the Fourier image has a /3 symmetry. This means that the PSF d oes n ot change if r otated by steps of /3. For example, diffraction along an y direction sh own as a solid line in Figure 2 is the same as alon g axis (in the following referred to as the A family). Along these directions, higher-order peaks are located equidistantly with radial steps of / 3 . The same is valid for any of the direction shown as a dash ed line in Figure 2 (referred to as the B famil y). These peaks have a regular separation of .

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Y



X



Figure 1. Segmented mirror. Segment number N=90, segmentation order M=5. Solid arrows illustrate the /3 symmetry.

Figure 2. Grid factor. The /3 symmetry is maintained in the Fourier plane: peaks are located equidistantly along an y solid (dashed) line.

Th e segment PSFs is the square modulus of the segment amplitude function, t (w ) , which for hexagonal segments is
t (w ) = = 1 2 ( ) exp i w d 2 A z sin 3 - sinc

(

)(

3 + + sin 3 + sinc 2 3

)

(

)(

3 -

)

(6) ,

with and given by Eq. 4. The Eq. 6 is important for all following analysis since it allows to d erive the exact analytical expr ession for the Strehl ratio and relative intensity of higher-order diffraction peaks associated with gaps, segment edge misfigure and tip-tilt error.

3. ANALYSIS OF DIFFRACTION EFFECTS
3. 1. Gaps Gaps are introduced by a shrinking the segment size from its ideal size given by the plane geometry of the hexagonally packed array. Con sider the case when the segment is a perfect hexagon with its center at the ideal position, but where flat-to-flat width, d, is slightly smaller than the center-to-center distance, d. For analytical con venien ce we introduce , the relative gap size: d - d (7) = = 1- d d . d If we assume segments t o be identical in size and shape, then the area of the segment A and the transmission function ( ) in Eq. 1 are independent of the index j. Th at allows to express the PSF as a product of the GF and the segment PSFs:
AN PSF (w ) = GF (w ) PS FS ( w , ) . z
2

(8)

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While the GF is still defined by Eq.3, the segment PSFs has become:
PSF
S

(w ) = (1

-

)2

1 2 ( )exp i w d 2 . A z

2

(9)

This function is defined only by the shape of an individual segment. As we will see, this approach is convenient to describe different diffraction effects associated with the strongly segmented mirrors. As the GF is equal to unit y at the center of the image coordinate system and in all points whose coordinates obey Eq. 5, for the calculation of the Strehl ratio and the intensity of higher-order diffraction peaks we need only to calculate the value of function PSFS(w) in these coordinates. From Eq. 9 we obtain the expression for the Strehl ratio, defined as the ratio of central peak intensity with gaps to the same intensity without gaps: 4 (10) S = (1 - ) 1 - 4 + 6 2 , i.e. the loss of the Strehl ratio is ­ unsurprisingly ­ equal to the square of the ratio (gap total area/ aperture area). Assu ming 1. 5 m segments with gaps of 12 mm (plausible value for OWL), the relative gap size is = 0. 008, and the Stehl ratio equals to 0.98. This relatively small ener gy loss is accompanied, however, by the appearance of a diffraction pattern, which could possibly complicate the scientific exploitation of data. Th e location of the GF peaks is defined by the distance between segment centers and does n ot chan ge with gap size. With an increase of gap size (decrease of segment size), the PSFS widens, its zeroes move outwards, and the GF peaks are revealed (Figure 3b). In the following, all peaks except for the central one are referred to as higher -order peaks.

(a)

(b)

Figure 3. Grid factor and segment PSF for a perfect mirror without gaps (a), and for a mirror with gaps between segments (b).

We ar e now interested in the ratio of high er-order peak to th e central peak intensity. From n ow on, we refer to this ratio as the higher-order peak relati ve intensity.

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Due to the symmetr y, peaks located at a fixed angular distance from the center have equal intensity. Figure 4 shows all peaks out to 0. 32" according to their angular distance from t he center (segment size 1.5m flat-t o-flat). They are label ed with a family name and an order number. Each family of peaks can be defined by its initial angle from horizontal: Famil y A: n ( 3) ; Famil y B: 6 + n ( 3) ;

Famil y C: arctan 3 5 + n ( 3) and arctan 3 2 + n ( 3) ; Famil y D: arct

( an(

3

) 7)

+ n ( 3) and arct

() an(3 3 2)

+ n ( 3) .

Th e number of possible families is infinite; in the following we concentrate only on those represented within a 0. 32" radius.

A1
0.08" 0.14"

B1 A
2

0.16"

0.21" 0.24"

C1

0.29" 0.32"

0.28"

A D
1

3

A

B2

4

Figure 4. Symmetry in the diffraction pattern: all peaks belongin g to the same circle have equal intensity. Segment size d=1.5m, =0.5 µm.

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10 H igher-o rd er peaks rel ati v e i nt ens ity

-3

10

-4

1

2, 6 7 8

4

Figure 5. Relative intensity of high er-ord er peaks as a function of gap size, = 0.5 µm, d=1.5m. Th e curve index corresponds to the group of peaks at angles: 1: 2: 3: 4: 5: 6: 7: 8:
24

3

10

-5

10

-6

10

-7

5

0. 0. 0. 0. 0. 0. 0. 0.

08 14 16 21 24 28 29 32



(A1) (B1) (A2) (C1) (A3) (B2) (D1) (A4)

10

-8

0

4

8

12 16 gap size, mm

20

Th e higher-order peak relati ve intensity is plotted as a function of gap size in Figure 5. Although all the curves are mon oton ous within the plotted range, peaks at larger angles are not always weaker than peaks located closer to the center. F or example, the series of peaks to be found at 0. 16 (A2) are fainter than peaks located at 0.21 (C1). A more thorou gh discussion of these phen omena will be given in a future paper. However, it is important to note that the structure of higher-order peaks in a h exagonal segmentation geometry is very different fr om that of a square segmentation geometry. In the latter case, a single family of peaks is observed, repeated alon g two orthogon al directions. In cl ose analogy with the simple 1D case (classi cal diffraction grating), the peaks are of similar amplitude out to an angular separation of about /(d). In the hexagon al case, such a beh avior is observed for the B family peaks as can be seen in Figure 5 where curves 2 (B1) and 6 (B2) are coincident. Th e brightest peaks are the six A1 peaks at 0. 08" from the center. The relative intensity of these peaks for small is

I A1 () 0 .7 2 .
Taking as an average for OWL 12mm gap and 1. 5m segments, we obtain: IA1(w=0.008)= 4· 10-5.

(11)

It sh ould be noted, however, that this figure is quite conservative. With all identical segments, OWL has gap size varying from ~6 mm to ~18 mm, and the projected geometry of the primary mirror is slightly irregular. As a result the higher-order peaks will be blurred, in addition to the effect of pu pil rotation.

3. 2. Turned Down/Up Edge We refer to turned d own/up edges to d escribe residual polishing errors in the form of phase profile misfigure appearing on the ed ge of the segments (Fig. 6). We introduce two i mportant parameters: width of misfigure , and depth . To mod el this effect, different functions for the edge profile can be used. On the basis of practical experience we assu me a quadratic function: 2 (12) ( x, , ) = -(x ) . Point x=0 corresponds to the "beginning" of the misfigure, x= to the edge of the segment.

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0



d
-



Figure 6. Turned down edge with a quadratic phase pr ofile on the segment edge. We assu me equivalent ed ge sh ape for all segments; the PSF can be presented as the product of the grid factor GF, introduced in Eq. 3, and the modified segment PSF, PSFs:

AN (13) PSF (w ) = GF ( w ) PS FS ( w , , ) . z By analogy with effects of gaps, to calculate the Strehl factor and the intensity of higher-order maxima, we have to deri ve an expr ession for the PSFs. The detailed calculation technique will be presented in a future paper. Here we present the final expression for the PSFs:
2 d - 2 + 2x d - 2 PSFS (w, , ) = t (w,0, ) + exp[i(x, , )] d x d 0 In this expression [F( x)]/x is the partial derivative of F(x) over x and (x, , edge (Eq. 12). The segment amplitude function t (w, x, ) is given by Eq. 6 with

2



2 (14) t(w, x, ) dx . ) is the phase profile at the segment

2

= [ (d - 2 + 2x ) 2z ]w

x

and = [ (d - 2 + 2x ) 2z ]w y .

(15)

The amplitude function t (w, x, ) is equal to unity at point w=0 for all x. For the Strehl ratio we obtain:

d - 2 + 2 x d - 2 (16) S( , ) = dx . + exp [i (x , , )] 4 d2 d 0 The exact analytical expression for the Strehl ratio can be evaluated assuming the quadratic phase function. The curves as a function of are shown in Fig. 7 for =5, 10 and 20mm, d=1.5m. Note that the curves saturate to (1- 2/d)4 . The saturation level of the Strehl ratio does not depend on the segment edge shape. In this limit, the integrand of the second term in Eq. 16 is a fast oscillating function, the integral of which for finite limits tends to zero. For any (x, , ) the saturation level is equal to the ratio [(d - 2 ) d ]4 . That is not surprising: infinitely deep slopes play the same role as amplitude gaps with a relative size =2/d, and the Strehl ratio in that limit is described by Eq. 10. For any given , the Strehl ratio does not decrease beyond its saturation level and we could use this level as a lower limit of S( , ) . However, this is arguably over-conservative. In practice the depth is not expected to exceed ~2 radians. The minimum of S( , ) in this range gi ves us a convenient estimate for the Strehl ratio, especially because the position of the minimum does not depend on :
2



2

2 2 S( , ) > 1 - 3.3 + 4.5 . d d
This gi ves 0.98, 0.95 and 0.92 for =5, 10 and 20mm correspondingly.

2

(17)

In the case of OWL, surface edge misfigure is expected to be in the range of a quarter wave length and to extend over 20 -10mm of a segment area (we assume the stress ­ free pol ishing on planetary polishers). This leads to an expected Strehl ratio in a range 0.94 - 0.97.

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6x 10

-4

1.00
5x 10
-4

0.98

H ighe r - O r d e r pe ak i n t e n s i ty

1

3
-4

4x 10

Strehl r a ti o

0.96

2

3x 10

-4

0.94

2x 10

-4

0.92 3 0.90

2
1x 10
-4

1
0

0

1

2 3 4 5 6 m is figure depth , w av e length uni ts

7

8

0

1

2 3 4 5 6 7 mis figur e depth , w a ve length uni ts

8

Figure 7. Strehl ratio for turned down edges as a function of misfigure depth. The curve parameter is misfigure width. 1: =5mm; 2: =10mm; 3: =20mm. Segment size d=1.5m. Dashed lines are the saturation level (12/d)4.

Figure 8. Relative intensity of higher-order peaks A1. 1: =5mm; 2: =10mm; 3: =20mm. Segment size d=1.5m. Dashed lines are saturation level 0.7·(2/d)2.

The intensity of the higher-order peaks is gi ven by the PSF'S of Eq. 14, where t (w , x, ) is calculated using points with coordinates from Eq. 5. We present here onl y the final expression for the bri ghtest A1 peaks. In Fig.6 there are six peaks located at 0.08" from the center:
2 2 I A 1 ( , ) sinc 2 1 - d 3 where C and S are the Fresnel integrals:

()

()

C + C 2 2

()

[ ( )+ S ( )]
2 2



,

(18)

C (x ) =

2


0

x

cos t 2 dt ;

()

S (x ) =

2 sin t 2 dt . 0

x

()

(19)

Eq. 18 is valid for any parameter . Again, this is an approximation to (/d)2 accuracy. The curves for =5, 10 and 20mm and d=1.5m are shown in Fig. 8. As we did above for the Strehl ratio we estimate the upper level of I1 within a range =0...2 by the value of its maximum, whose position in this case is independent of . So the intensity of these peaks is no higher than
2 . I A1 ( , ) < 0 .5 d For =5, 10 and 20mm the estimation is 2·10-5, 8.5·10-5 and 4·10-4 correspondingly.
2

(20)

These estimates shall be deemed as very pessi mistic, as the expected surface misfigure for OWL segment edges should be of ¼ wavelength. Then the expected values for I1 peaks relative intensity is 7·10-5 - 3·10-4.

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3.3. Tip-Tilt Effect A detailed analysis of diffraction effects associated with tip-tilt errors for mirrors with a large number of segments can be found elsewhere.1 Here we present the main results of the study because in many respects they are similar to that caused by gaps or turned up/down edges. An aperture with random segments tip-tilt errors behaves as a randomly blazed diffraction grating. The loss of intensity in the central peak is accompanied by the appearance of regular diffraction peaks around the central one on a background of random speckles. The PSF consists of two terms: the first item describes the regular diffraction pattern; the second item describes the non-regular speckle field. The former is nearl y independent of segment number, although the intensity of the non-regular component decreases as N-1. For a small number of segments the regular pattern "is lost" in speckles, but for large N it dominates. As the main object of the present paper is the diffraction effects associated to strong segmentation, we assume that the non-regular component is negligible and in the following we concentrate on the regular term. As in Eqs. 8 and 13 the ensemble averaged PSF (its regular part) is a multiplication of the grid function GF and a modified PSF for one segment:

AN (21) GF (w ) PS FS (w , rms ) . z The GF is again the grid factor from Eq. 3. The modified PSFS (w, rms) depends on the tip-tilt rms. So we observe the analogy with the modified segment PSF from Eqs. 9 and 14. The following analysis is directly analogous with those presented in subsections 3.1 and 3.2. PSF (w , rms

)

2

The modified PSFS (w, rms ) was found to be

1

PS Fs ( w , rms ) =



t ( w )Q (rms , w - w )d 2 w ,
2

(22)

where the function t( w) is the segment amplitude function (6), and Q (rms, w - w ) is a Gaussian function with a width equal to 2.7rms:
2 2 2 (w - w )2 d d2 2 Q (rms , w - w ) = exp - 2 z 2 ( 2.7 rms ) z 2 ( 2 .7 rms ) 2 2

.

(23)

The ensemble averaged Strehl ratio is the value of PS FS (w , rms ) for the point w=0. The curves are presented in Fig. 9 for N=37, 217, and 817. For a small tip-tilt rms value the Strehl ratio can be approximated as the following: rms 4 2 (24) 2.34 + . 4 N To the second order, this expr ession coincides with the Marechal approximation. Note the weak dependence on number of segments which is in contrast to the case of pure piston errors2,6 where the Strehl ratio reaches a floor level equal to 1/N. S(rms ) 1 - rms 2 +

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1

10
S t rehl ra tio

0

1 2

H ighe r-o rder peak re la tiv e in tens ity

3 4

10

-1

10

-4

5

6 7 8
-8

10

-2

10

10

-3

1 2 3

0. 0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

10

-1 2

W av ef ron t t ip -t ilt error rm s , w av eleng th units

0.0

0. 2 0.4 0. 6 0.8 W a v efr on t tip- tilt e rr o r r m s , w av e len gth u nits

Fig. 9. Strehl ratio as a function of tip-tilt rms for different numbers of segments: 1: N=37; 2: N=217; 3: N= 817.

Fig. 10. Relative intensity of higher-order peaks as a function of tip-tilt rms. N=217. The curve index corresponds to the group of peaks at a definite angle: 1: 0.08(A1); 2: 0.14(B1); 3: 0.16(A2); 4: 0.21(C1); 5: 0.24(A3); 6: 0.28(B2); 7: 0.29(D1); 8: 0.32(A4).

The position of higher-order peaks is defined by the GF and their value can be calculated from the PS FS (w , rms ) using points from Eq. 5. Unlike the effects considered above, the intensity of peaks with the same index and order are not equal anymore. For example, there is a slight difference in the intensities of the six A1 peaks for a short exposure image. We can create analytical expressions onl y for ensemble averaged characteristics. A more precise statistical analysis can be found elsewhere.1 The ensemble averaged relative intensity of higher-order diffraction peaks is shown in Fig. 10 for N=217. Li ke the Strehl ratio, the intensity of the peaks for large N is insensitive to the number of segments. The curve saturation is the result of normalization: for large rms the higher-order peaks decrease according to the same law as the Strehl ratio. As we did above, we present here the approximate expression for the averaged intensity of A1 peaks for the small rms:
I A1 (rms ) 0.013rms 4 . Taking as a reason able estimate for OWL a tip-tilt rms=/30, we obtain S=0.96 and IA1 =2.5·10-5. (25)

3.4. Piston Effect
Although the main intention of the present paper is to enclose all effects which lead to the appearances of the diffraction pattern, the piston effect should not be set aside. Th e PSF for the segmented aperture with random piston distribution can be represented as a product of the GF and segmented PSF. But unlike in all previous cases, the presence of piston errors does not chan ge the segment PSFs, but disturbs the periodic structure of the function GF. As the result in a whole PSF the n oisy undergrowth of speckles appears, but not the diffraction pattern of the regular peaks.

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Th e appearance of speckle is accompanied by a reduction in the Strehl ratio. The expression for the ensemble averaged Strehl ratio is2,6 1 (26) S= 1 + (N - 1) exp (- rms 2 ) , N Note the saturation of the Strehl ratio to the level N-1.

[

]

Th e an gular size of speckle field does n ot depend on the piston rms and can be defined as a full width at half maximu m of the PSFs : (27) speckle 2.9 d . For =0.5 µm and d=1.5m sp
eckle

~0.07.

Th e ratio R (averaged speckle/central peak intensity) can be found by appl ying energy conser vation and the assumption that the average width of an each individual speckle is the same as the width of the central peak. If we use the fact that the number of speckles is equal to the number of segment N then we obtain: 1 - exp( -rms 2 ) . (28) 1 + (1 - N )exp(-rms 2 ) Th e OWL expected piston rms is /30, that gives 0.96 for the Strehl ratio and R = 3·10-5 for the relative intensity of the speckled field.
R=

4. RESULTS
We have presented a unified approach to the study of diffraction effects from gaps, segment ed ge misfigure, and random piston and tip-tilt errors. All considered effects lead to a loss of intensit y in the central point and a disturbance of the PSF structure. Gaps and segment edge misfigure produce the regular pattern of the higher-order diffraction peaks, piston errors produ ce only the speckles, and tip-tilt errors ­ the combination of speckles and the higher-order peaks. For each effect, the telescope PSF can be represented as a produ ct of the GF and the segment PSFs, modified specifically for each case. Th e Strehl ratio and the relative intensity of diffraction peaks can be found by calculating the segment PSFs at the points where the peaks of the GF are located. For the Strehl ratio and the relative intensity of the six bri ghtest peaks (A 1), approximate expressions h ave been obtained. We su mmarize the results in Table 1, in which numerical values for typical parameters are also sh own.

Table 1. Strehl ratio and relative intensity of the first order diffraction peaks. Critical paramet ers
gap size, segment size d Turned edges Tip-tilt depth, width, rms segments, N rms segments, N

Effect

Typical value
12mm 1.5m 0.25µm (WF) 20 -10mm /30 (WF) 3300 /30 (WF) 3300

Strehl ratio Expression
(1-/d)
4

Value
0.98 0.94 0.97 0.96

Relative intensity of A1 peaks Expressi on Value
0.7(/d)
2

Gap

4·10

-5

> 1-3.3·(2/d) + 4.5·(2 /d)2 1-rms
2

< 0 . 5 ( /d )

2

7·10-5 3·10
-4

+ rms4(2.34+2/N)/4 exp( -rms2)

0.013rms4

2.5·10

-5

Piston

0.96

__

__

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5. CONCLUSION
Th e next generation of large telescopes will have highly segmented apertures. It is therefore i mportant to study the optical effects caused by such segmentation. The four effects (gaps, segment edges misfigure, and random piston and tip-tilt errors) examined here are susceptible to analytical methods, creating the groundwork for accurate simulations. Studying gaps, edges misfigure, and random tip-tilt errors we have con cent and the relative intensity of higher-order diffraction peaks. For the l distinguishes the peaks of different diffraction orders and takes into account provide a useful insight into the performan ce of highly segmented mirrors. designing the next gen eration of large telescopes. rated atter the / Takin on two main aspects: the Strehl ratio we suggest a classification, which 3 symmetry. The derived expressi ons g into account these effects will help

ACKNOWLEDGMENTS
Th e authors wish to acknowledge that this research is supported by th e European Commissi on RTN program: "Adaptive Optics for the Extremely Large Telescopes", under contract #HPRN -CT-2000-00147.

REFERENCES
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*n yaitsko@eso.or g, phone +49 893200 6581; fax +49 89 320 2362, www.eso.org, ESO, Karl-Schwarzschild-Str. 2 D85748 Garching bei MÝnchen.

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