Analytical expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT

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1 ASTRONOMY & ASTROPHYSICS MAY II 000, PAGE 57 SUPPLEMENT SERIES Astron. Astrophys. Suppl. Ser. 44, ) Analytical expressions or ield astigmatism in decentered two mirror telescopes and application to the collimation o the ESO VLT L. Noethe and S. Guisard European Southern Observatory, Karl-Schwarzschild-Str., Garching, Germany lnoethe@eso.org European Southern Observatory, Casilla 900, Santiago 9, Chile sguisard@eso.org Received January 7; accepted March 9, 000 Abstract. We derive ormulae or all parameters deining the ield astigmatism o misaligned two mirror telescopes with arbitrary geometries and with stop positions anywhere on the line connecting the vertices o the two mirrors. The ormulae show explicitly the dependence o the ield astigmatism on the undamental design parameters and characteristics o the telescope and on the stop position. Special attention is given to the particular case where such a schiespiegler has been corrected or coma at the ield center. In addition, we study the eects o the practical deinition that the center o the ield is the center o the adapter. Following a recent paper by McLeod, where the ield dependence o astigmatism is used to collimate a Ritchey-Chretien telescope with the stop at the primary mirror, we apply our ormulae to the Cassegrain ocus o the ESO Very Large Telescope VLT), where the stop is at the secondary mirror and the telescope is only corrected or spherical aberration. We present measurements o the ield astigmatism and discuss the accuracy o the collimation method. Key words: telescopes. Introduction A general theory o low order ield aberrations o decentered optical systems has been given by Shack & Thompson 980). In particular it has been shown that the general ield dependence o third order astigmatism can be described by a binodal pattern, known as ovals o Cassini. Only or special cases such as a centered system do the two nodes coincide and the ield dependence reduces to the well known rotationally symmetric pattern with a quadratic dependence on the distance to the ield center. These general geometric properties have been used by McLeod 996), starting rom equations by Schroeder 987), or the alignment o an aplanatic two mirror telescope. We shall recall this method briely. In a Cassegrain telescope the absence o decentering coma in the center o the ield does not imply that the optical axes o the primary M) and the secondary mirrors M) coincide. The axes o the primary and the secondary mirrors must intersect at the coma ree point, but the axis o M may still orm an angle α with respect to the axis o M. For this case McLeod showed that the components Z 4 and Z 5 o third order astigmatism o a two mirror telescope with the stop at the primary mirror or a ield angle θ with components θ x and θ y are given by Z sys 4 = B 0 θx θ y )+B θ x α x θ y α y ) + B α x α y ) ) Z sys 5 =B 0 θ x θ y + B θ x α y + θ y α x )+B α x α y. ) B 0 is the coeicient o ield astigmatism or a centered telescope, B and B only appear in decentered systems. Numerical values or B 0, B and B were obtained by using general ormulae or ield astigmatism o individual mirrors and adding the eects o the two mirrors. The values or α x and α y could then be obtained rom measurements o Z sys 4 and Z sys 5 in the ield o the telescope. For a centered two mirror telescope Wilson 996) has derived expressions or the low order ield aberrations including third order astigmatism B 0 ) showing the dependence on undamental design parameters and optical properties o the total telescope and on the position o the stop along the optical axis. This gives more physical insight into the characteristics o ield aberrations o two mirror telescopes. Similarly we derive, or decentered two mirror telescopes, explicit expressions or third order astigmatism, i.e. or the parameters B and B,and

2 58 L. Noethe and S. Guisard: Field astigmatism in decentered telescopes discuss the ield dependence o astigmatism or undamental types o two mirror telescopes. In a centered optical system the ield center, projected towards the sky, is the direction parallel to the optical axis o M. The ormulae or the parameters B 0, B and B are initially derived or this reerence system. However, in a decentered system the image o an object in the ield center is not in the mechanical center o the adapter, where the instruments are located. For all measurements in the telescope, the practical ield center is the center o the adapter. The evaluation o the measurements o ield astigmatism has to take this dierence o the origins o the reerence systems into account. The structure o the Eqs. ) and ) will remain the same, but the parameters B 0, B and B will change and θ will denote the ield angle with respect to the center o the adapter. One case o practical interest is the collimation at the Cassegrain ocus o the VLT where the stop is at the secondary mirror and the telescope is only corrected or spherical aberration. We apply our ormulae to measurements o astigmatism in the ield o the Cassegrain ocus to determine the misalignment angles α x and α y. In addition, the possibility to change the misalignment angles by well deined values by rotating the primary mirror around its vertex allows a measurement o the accuracy o this collimation method.. Astigmatism in a misaligned telescope with an arbitrary pupil position.. General ormulation The general case which we study is shown in Fig.. The notation as well as the sign conventions or the angles and the distances are taken rom Wilson 996). The ollowing list deines the parameters. i = stands or the primary mirror and i = or the secondary mirror. y i : semi-diameter o the mirror i i : ocal length o the mirror i s pri : distance rom the surace to the entrance pupil o mirror i b si : aspheric constant o the mirror i n i : index o the exit medium o mirror i u pri : angle o the incoming principle ray with the axis o the mirror i h i : distance rom the axis o the mirror i to the center o its entrance pupil z : distance o the coma-ree point rom the surace o M. In this whole section the ield center is deined as the optical axis o M, that is u pr = 0. The stop is located between the two mirrors at a distance s pr rom M and decentered laterally by a value h rom the M axis and h rom the M axis. The entrance pupil is located at a distance s pr behind M and decentered by h rom the M axis. The vertex o the secondary mirror is decentered by a distance δ rom the M axis and tilted by an angle α with respect to the M axis. Initially we will only make the assumption that the lateral decenter δ and the rotation o M around its vertex by α are in the same plane. Ater a bit o geometry ollowing Fig. we ind: ) spr u pr = u pr α h 3) s pr h = h 4) d + s pr h = h δ s pr α. 5) Knowing the distance s pr rom M to the stop in a plane between M and M one can calculate s pr by s pr = s pr + d s pr + d. 6) Inversely, s pr is given by s pr = s pr s pr d. 7) The coeicient o the astigmatic waveront aberration o a two mirror telescope is then given by Schroeder 987, page 79) c ast = D y + D y 8) D i = n i A0,i u 6 3 pri + A,iu pri h i + A,i h ) i 9) i with the expressions or A 0,i given by Schroeder 987) in the Tables 5.6 and 5.9 but adding the missing actor / in the expressions or astigmatism) A 0,i = b si s pri + i s pri) 0) A,i =4 i + b si )s pri ) A,i =+b si. ) Introducing the expressions or A 0,i, A,i and A,i into Eq. 9) one gets D i = n i {[b 6 3 si s pri + i s pri) ] u pri i +[4 i + b si)s pri ] u pri h i + + b si ) h i } 3) Our deinition o the waveront error diers by a actor rom the one given by Schroeder 987), since we use the convention that a waveront error is positive i the actual waveront is in advance o the reerence waveront... Explicit expressions or astigmatism in a schiespiegler o general orm... Telescope in ocal orm In a two mirror telescope as shown in Fig. the horizontal position o the stop may be anywhere to the let o the primary mirror. For the vertical position h o the stop we make the simpliying assumption that the center o the

3 L. Noethe and S. Guisard: Field astigmatism in decentered telescopes 59 M M C α P cp δ u pr h h h z u pr s pr Stop s pr Entrance pupil d Fig.. Two mirror telescope with general position o M and general position o the stop stop lies on the line connecting the vertices o the two mirrors. h is then given by h = δ d d + s pr ) = δ s pr d s pr. 4) With this expression or h one gets or the Eqs. 3), 4) and 5) u pr = s pr u pr α + δ d s pr 5) h = δ s pr 6) d ) δ spr ) h = α d s pr d. 7) Introducing these expressions into Eqs. 3) and 8) and using the relationships given by Wilson 996, Sect...5.) m = d 8) = m 9) = L m + 0) L = m d ) ) y = L y ) where m is the magniication o the secondary mirror, the ocal length o the two mirror telescope and L is the distance rom the secondary mirror to the ocus o the telescope, one gets with n = andn =orthe coeicients o the astigmatic waveront aberration c ast = C 0 u pr + C u pr + C 3) C 0 = ) { y 4 L + d ) + d L ξ + s pr +d ξ) ) } spr + ζ + Lξ) 4) ) y C = 4 { [ α m +) + L + d ) s pr + δ [m +) + d L m + ) + b s ) d ξ + + s pr d ) y ] Lm ) ) d ) 4 m +) [m m +)b s ] ) ]} spr + ζ + Lξ) d C = 4 { α Lm +) + αδ [ m +) s ] pr L m ) d 5)

4 60 L. Noethe and S. Guisard: Field astigmatism in decentered telescopes where [ + δ 4L m +) 3 + b s ) s pr m +) [m m +)b s ] d ) ]} spr + d ζ + Lξ) 6) ζ = m3 4 + b s) 7) [ ξ = m m +) 3 ) + b s]. 8) 4 m + The expression or C 0 is, apart rom a actor needed or the conversion to Seidel coeicients, identical to the one given by Wilson 996, Sect ). I the stop is at the primary mirror, the parameters C 0, C and C should not depend on the asphericity b s o the primary mirror. This can be seen rom the Eqs. 4), 5) and 6), which then depend only on the asphericity b s o the secondary mirror. Similarly, i the stop is at the secondary mirror, the parameters C 0, C and C should not depend on the asphericity o the secondary mirror. This can easily be veriied by introducing s pr = 0 or, equivalently, rom Eq. 6), s pr / = d /L into the Eqs. 4), 5) and 6). These expressions can be considerably simpliied by using corresponding expressions or other aberration coeicients o two mirror telescopes Wilson 996, Sects and 3.7.): c spher o spherical aberration, c coma,cen o third order ield coma o a centered system, and the coeicients o ield independent third order coma generated by a lateral decenter by δ c coma,δ ) and by a pure rotation o M around its vertex by α c coma,α ). c spher = ) 4 y 8 ζ + Lξ) 9) ) 3 [ d ξ y C,0 = ) [ y 4 α m +) + L + d ) + δ m +) + d )] L m + ) + b s ) 35) C,0 = ) [ y 4 α Lm +) ] + αδ m +) +δ 4L m +) 3 + b s ). 36) Equations 4), 5) and 6) can be written as C 0 = C 0,0 s ) pr c coma,cen y spr c spher 37) y C = C,0 s ) pr δ c coma,cen y d + c coma,α + c coma,δ ) spr + 4 δ c spher 38) y d C = C,0 s pr δ c coma,α + c coma,δ ) y d ) ) spr δ + 6 c spher. 39) y d These equations show a nice symmetry o the stop-shit terms. The linear terms are proportional to coeicients o coma and the quadratic terms are proportional to the coeicient o spherical aberration. The total coma in the linear coeicient o C is the sum o the coma or a centered system or a principal ray with the angle δ/d,which,in the decentered system, is the angle o the principal ray connecting the vertices o the two mirrors, and the coma contributions rom a pure decenter o M by δ and a pure rotation o M around its vertex by α. The total coma in the linear coeicient o C contains only the two latter contributions. c coma,cen = s ] pr ζ + Lξ) u pr 30) ) 3 y c coma,δ = 4 [ m +) [m m +)b s ] + s ] pr ζ + Lξ) δ 3) d c coma,α = ) 3 y 4 Lm ) α. 3) With the urther deinitions c coma,cen = c coma,cen u pr 33) C 0,0 = 4 y ) [ ] L + d )+ d L ξ 34)... Telescope in aocal orm Equations 9) to 39) can be converted into equations valid or aocal telescopes, where both the total ocal length and the position o the ocus, which is linked to L, go to ininity. One can eliminate and L in avour o, d and m with the Eqs. 9) and ) and then let m go to ininity. This gives c spher,a = ) 4 [ y 3 + b s ) ] + d ) + b s ) 40) c coma,cen,a [ = 8 y +4 s pr c spher,a y ) 3 d + b s) ] u pr 4)

5 L. Noethe and S. Guisard: Field astigmatism in decentered telescopes 6 [ ) 3 y c coma,δ,a = 8 b s) + s ] pr 4 c spher,a δ 4) y d c coma,α,a = ) 3 y 4 d ) α 43) C 0,0,a = ) y d 6 d + b s ) 44) C,0,a = ) [ y 4 α d ) )] d + δ + d ) + b s) 45) C,0,a = ) [ y 4 α d )+ αδ ] + δ +b s 4 d. 46) ) Equations 37), 38) and 39) or ocal telescopes are then also valid or aocal telescopes i all expressions in these ormulae are replaced by the corresponding expressions 40) to 46) or aocal telescopes..3. Explicit expressions or astigmatism in a coma-ree schiespiegler.3.. Telescope in ocal orm A urther simpliication is possible i the telescope is corrected or coma at the center o the ield, that is or u pr = 0. The axes o M and M must then intersect at the coma-ree point P cp. The distance rom the vertex o M to P cp is denoted by z. The lateral decenter δ and the misalignment angle α are then related by δ = αz. z can be calculated rom the requirement that the contributions to decentering coma rom a pure lateral decenter δ and the simultaneous rotation α = δ/z cancel. c coma,δ + c coma,α =0. 47) This gives, using the Eqs. 3) and 3), z =L m m + 48) m m +)b s spr 4 m +) d ζ + Lξ) This equation shows that the position o the coma-ree point depends on the stop position. I the stop is at the primary mirror or i the telescope is corrected or spherical aberration, z depends, or a given telescope geometry L and m, only on the aspheric constant b s o the secondary mirror. z =L m 49) m + m m +)b s I the stopisatthe secondarymirrorz depends, or a given telescope geometry L and m, only on the asphericity b s o the primary mirror. m z =L m m ) L m3 + b s) 50) The coeicient c ast o third order astigmatism can then be expressed as c ast = B 0 u pr + B u pr α + B α 5) with B 0 = C 0 5) B = ) [ y z d m +) 3 4 d L m b s +d ξ + s pr + L +m )] +)d ζ + Lξ) m )L B = ) ) [ ) y z d m + 4 b s ξ d L m + s ) pr d m + b s ζ + Lξ) L m ) spr + ζ + Lξ) 4 m ) m +) ζ + Lξ L 53) )].54) Equations 5), 53) and 54) show that the parameters B i are proportional to z/d ) i. At a irst glance they seem to be linear or quadratic equations in s pr /. This is only the case or B 0 since the distance z o the coma-ree point appearing in B and B depends itsel on the stop position, as can be seen rom Eq. 49). By introducing z = δ/α in the Eqs. 5), 53) and 54) the coeicient o third order astigmatism can be expressed as a polynomial in u pr and δ, thatis c ast = B 0 u pr + Bδ) u prδ + B δ) δ. 55) It is easy to see that now the parameter B δ) is a linear unction and the parameters B 0 and B δ) are quadratic unctions in s pr. Exactly as with C 0, C and C the parameters B 0, B and B depend only on the asphericity b s o M, i the stop is at the primary mirror, and only on the asphericity b s o M, i the stop is at the secondary mirror. Since the coupling between α and δ through z involves the expression or spherical aberration, the symmetry o the stop-shit terms has disappeared. But, the telescope will usually be corrected or spherical aberration. In this case all terms containing ζ + Lξ vanish. Then, with the additional deinitions B 0,0 = ) [ ] y 4 L + d )+ d L ξ 56)

6 6 L. Noethe and S. Guisard: Field astigmatism in decentered telescopes B,0 = ) y d m +) 3 4 L m b s 57) B,0 = ) y d ) m + 4 b s ξ. 58) L m Equations 5), 53) and 54) reduce to B 0 = B 0,0 s pr c coma,cen 59) y B = z B,0 + s ) pr 60) d B = y c coma,cen ) z B,0. 6) d The distance z rom the secondary mirror to the comaree point and the expression c coma,cen are now no longer dependent on the stop position. Thereore, B 0 and B are linear unctions o the stop position and B is independent o the stop position..3.. Telescope in aocal orm The Eqs. 59), 60) and 6), all valid or ocal telescopes corrected or spherical aberration, can be converted into equations valid or aocal telescopes, also corrected or spherical aberration, by using Eqs. ) and 9) and letting m go to ininity. The distance rom M to the comaree point is then z = d b s. 6) For a classical or aplanatic aocal telescope the coma-ree point is in the ocus o M. The astigmatism parameters are given by B 0 = ) [ y d ] d 4 + b s) 4 d + s pr 63) B = ) [ y 4 b b s s + s d ] pr + b s ) 64) y B = ) +b s 4 b s ) b s d ). 65) For a Mersenne telescope with b s = b s = oneobtains immediately B 0 = B = 0. This gives thereore pure linear astigmatism which is proportional to the misalignment angle α with the center o the pattern at the center o the ield. This is a nice example o the general statement by Shack & Thompson 980) that a system which is ree o astigmatism in the centered coniguration will show either linear or constant astigmatism in the decentered coniguration..4. Conclusions or speciic types o telescopes and stop positions.5. Speciic stop positions at coma-ree schiespieglers For a general coma-ree schiespiegler the ollowing conclusions can be drawn or the stop positions at the primary and secondary mirrors. Stop at the primary mirror For a spherical secondary b s = 0) the Eqs. 53) and 54) show that one has, as or a centered system, B = B = 0. Because o z = the coma-ree point is at the center o curvature o M. Since a rotation o a spherical mirror around its center o curvature does not change the optical characteristics o the telescope, the astigmatism pattern has to be the same as the one o the centered system. I ξ = 0, one has B = 0, that is one node is at the position o the image corresponding to the ield center. Stop at the secondary mirror I one uses the substitution s pr / = d /L in Eq. 54) one can show that B is proportional to ζ. Thereore, or a coma-ree schiespiegler with a parabolic primary mirror one node is at the position o the image corresponding to the ield center..6. Speciic types o coma-ree schiespieglers corrected or spherical aberration From the general ormulae given above more speciic conclusions can be drawn i the telescopes is corrected or spherical aberration. In all cases we will also discuss the results in the limit o large magniications, deined here as m together with a inite distance L rom the secondary mirror to the ocus. The semi-diameter y o M, its radius o curvature and the dierence between and d will go to zero. This is dierent rom the case o an aocal telescope, where L goes to ininity, while y, and d remain inite. For the limit case with inite L we get rom Eqs. 56) to 58) the ollowing expressions or the astigmatism parameters: Stop at M Stop at M B 0 + y 6 L + b s)m y 6 L + b s)m + y + y 4 L 4 L 66) B y b s y b s 4 B 0 0. We now discuss a ew telescope types. Classical telescope The primary mirror is parabolic and ξ = 0. Thereore B vanishes. The consequence o this is that one o

7 L. Noethe and S. Guisard: Field astigmatism in decentered telescopes 63 the two nodes is exactly at the image position or an object on the axis o the primary mirror, that is with u pr =0. For the limit o large magniications one has b s and thereore rom Eq. 66) independently o the stop position: y B 0 4 L B y 4 B 67) L B 0 Since the distance L between the vertex o M and the ocus will generally be similar to the ocal length o M the ratio B /B 0 will be close to. Ritchey-Chretien telescope All three parameters are independent o the stop position, since, in addition to spherical aberration, coma is zero as well. Since modern Ritchey-Chretien telescopes with large values o m are very close to classical telescopes, the conclusions drawn or classical telescopes will be approximately valid. That is, B will be much smaller than B 0 and B, a point also discussed by Wilson 999, Sect...), and the ratio B /B 0 will be close to. Telescope with a spherical secondary mirror Dall- Kirkham) Because o b s = 0 the parameter B is zero and B is only proportional to the coma o the telescope. One node is thereore at the image position corresponding to the ield center with u pr = 0. I, in addition, the stop is at the primary mirror, B is also zero. In that case the two nodes coincide and one has pure quadratic astigmatism with the center on the axis o the primary mirror. For the limit o large magniications one has rom Eq. 66): Stop at M Stop at M B 0 + y 6 L m y 6 L m B 0 B 4 y 68) 0 4 L B 0 Whereas B converges to values independent o m, B 0 is proportional to m. The binodal nature o the ield astigmatism diminishes thereore linearly with increasing magniication o the telescope. For a certain stop position between the primary and the secondary mirror the coeicient B 0 o quadratic astigmatism vanishes. For the VLT with the Nasmyth ocus the stop would be 3.9 m in ront o the secondary mirror. The coeicient o linear astigmatism would be small, B µm/deg, but the coeicient o ield coma would be very large, c coma,cen 660 µm/deg. Telescope with a spherical primary mirror For large magniications we have in the limit m b s m /L and then rom Eq. 66): Stop at M Stop at M B 0 y ) 6 L m y ) 6 L m y y 69) B 4 ) ) B L L 8 B 0 4 B converges to values independent o m. The limit values o B 0 are by a actor m /L larger than the ones or a telescopes with a spherical secondary. The binodal nature o the ield astigmatism diminishes thereore quadratically with increasing magniication o the telescope..7. Numerical examples or the VLT In this sections we give numerical examples or the Cassegrain ocus o the VLT telescope. The optical parameters used or these calculations are summarized in the ollowing table mm mm b s b s.6696 y mm y mm d mm z mm L mm m I the pupil was located at the primary mirror, we would have s pr = 0. This would give the ollowing numerical values or the parameters B 0, B and B. B 0 = µm/deg B = µm/deg B = µm/deg. 70) As has been discussed beore, the value o B is eectively negligible compared with the values o B 0 and B and the ratio B /B 0 is approximately equal to. In reality the pupil is located at the secondary mirror. Then s pr = 0 and rom Eq. 6) one gets s pr = mm. This gives the ollowing numerical values or the parameters: B 0 = µm/deg B = µm/deg B = µm/deg. 7) Since the spherical aberration is zero the value o B is, as can be seen rom Eq. 54), identical to the one or the stop at the primary mirror. But, there is a signiicant dierence between the values o B 0 orthetwostoppositions.

8 64 L. Noethe and S. Guisard: Field astigmatism in decentered telescopes C δ θ oset M L M Focus Fig.. Pure lateral decenter o M by δ. C is the center o curvature o M θ C α oset M L M Focus Fig. 3. Pure rotation o M by α around its vertex. C is the center o curvature o M 3. Recalculation o the coeicients B 0, B and B or the center o the adapter 3.. Introduction In the ormulae given above the center o the ield is deined by u pr = 0 and corresponds to the direction o a star located on the M axis. In a decentered telescope the rays coming rom such a star do not ocus in the center o the adapter. The center o the adapter is however always deined as being the center o the ield in a telescope and ield measurements are made with reerence to this point. In this section we irst calculate the oset angles o the telescope corresponding to the misalignment o M and then the eect o these osets on the parameters B 0, B and B. 3.. Depointing o the telescope or a pure lateral decentering o M Depointing or a pure lateral decentering o M I M is laterally decentered by δ see Fig. ) the star centered in the adapter is producing an angle θ oset,d with the M axis: θ oset,d = L δ. 7) For the VLT at the Cassegrain ocus the actor in ront o δ is /mm. Depointing or M tilted around its vertex I M is tilted by an angle α around its pole see Fig. 3), the star centered on the adapter is producing an angle θ oset,t with the M axis. θ oset,t = L α. 73) For the VLT at the Cassegrain ocus the actor in ront o α is Depointing or M rotated around the coma-ree point When M is rotated around its coma-ree point by an angle α we have a combination o pure tilt and lateral decentering with δ = αz. The resulting oset o the telescope is θ oset,cp with T = L = θ oset,t + θ oset,d = Tα 74) z ). 75) For the VLT at the Cassegrain ocus one has T = New coeicients B 0, B and B with reerence to the center o the adapter To get the coeicients B 0, B and B with the reerence o the ield in the center o the adapter, we deine new ield angles as θ = θ Tα. 76) Replacing then θ by θ + Tα in the Eqs. ) and ) leads to Z sys 4 = B 0 θ x θ y )+B θ xα x θ yα y ) + B α x α y) 77) Z sys 5 =B 0 θ x θ y + B θ x α y + θ y α x)+b α xα y. 78) Equations 77) and 78) are similar to Eqs. ) and ), but with dierent coeicients B 0, B and B. The new parameters B 0, B and B are given by B 0 = B 0 B = B +B 0 T B = B + B T + B 0 T. 79)

9 L. Noethe and S. Guisard: Field astigmatism in decentered telescopes 65 For the VLT with the pupil at M B 0 = µm/deg B 0 = µm/deg B = µm/deg B = µm/deg B = µm/deg B = +4.8 µm/deg. For the VLT with the pupil at M B 0 = µm/deg B 0 = µm/deg B = µm/deg B = µm/deg B = µm/deg B = µm/deg. In both cases the new coeicients B are approximately 30% larger than the original coeicients B Eects o the correction o coma at the center o the adapter The ormulae in Sect..3 have been derived under the assumption that the telescope was a coma-ree schiespiegler. This means that the telecope is corrected or coma or the ield center with u pr = 0. But, in reality the coma correction will ensure that the telescope is ree o coma at the center o the adapter. The ormulae derived above can thereore, strictly speaking, not be applied to non-aplanatic telescopes. But, a short calculation will show that, in practice, the inaccuracies introduced by this eect are negligible. Equation 74) gives the relationship between the misalignment angle and the corresponding ield angle. The coeicient o coma or this ield angle can then be calculated rom Eq. 30). Finally, one needs the coeicient c coma,coc o coma generated by a rotation around the center o curvature o M by an angle ϕ. This can be derived by combining the expressions or coeicients o coma generated by a pure lateral decentering by δ and a rotation around the vertex o M by an angle u pr see Eqs. 3) and 3)). I δ and u pr are related by δ = u pr the total eect is a rotation around the center o curvature o M. One then obtains c coma,coc = ) 3 y 4 m ) L Tϕ. 80) z Combining the three Eqs. 74), 30) and 80) gives the rotation angle ϕ o M which shits the coma correction rom the ield center with u pr = 0 to the center o the adapter. ϕ = d ξ + ) z m ) L α α d ξ + = m +)[m m +)b s ] 8) For a classical telescope this reduces to the simple expression ϕ class = α 8) m m ) Since the VLT is optically very close to a classical telescope and m = 7.556, ϕ is approximately ity times smaller than α. This shows that the dierence between a schiespiegler with coma corrected or the center o the ield and one with coma corrected or the center o the adapter is negligible. The ormulae or ield astigmatism derived above or a schiespiegler ree o coma at the center o the ield can thereore also be used or a schiespiegler with coma corrected or the center o the adapter Simulation or the ESO 3.6 m telescope As an example we checked our ormula with a simulation done with Zemax or the 3.6 m telescope on La Silla. We simulated M tilted around the coma-ree point by an angle α =0.36, which corresponds to an unusually large decenter o δ = 0 mm. The values o astigmatism calculated at dierent ield positions by Zemax were entered into our itting sotware. Two ittings were done, one with the set o parameters B 0, B, B or the original ield center and one with the set B 0, B and B or the ield center at the center o the adapter. With the irst set we ind α = with a residual waveront rms o 697. nm. With the second set we ind α =0.04, very close to the input value, with a small residual rms o 0.nm. 4. Misalignment measurements at the VLT 4.. General procedure The VLT has both Nasmyth and Cassegrain ocii. It is optimised to be a Ritchey-Chretien telescope at the Nasmyth ocus. The Cassegrain ocus has a dierent aperture ratio rom that o the Nasmyth ocus. To switch rom the Nasmyth to the Cassegrain ocus one has to change the distance between the mirrors done by reocussing with M) and bend the primary mirror to remove spherical aberration. In its Cassegrain coniguration the telescope is no longer aplanatic. Furthermore, the stop is at the secondary mirror. For the measurement o the misalignment using the eatures o the ield astigmatism one thereore has to use the constants B 0, B and B given at the end o Sect The measurement procedure is then as ollows. The telescope is irst corrected or coma at the center o the adapter. With B 0, B and B known, the values or α x and α y could, in principle, be obtained rom one measurement o Z sys 4 and Z sys 5 somewhere in the ield o the telescope. But, in large telescopes ield independent astigmatism can easily be generated elastically. In addition, the measurements are, or example due to local air eects, not ree o noise. Thereore, it will be necessary and more accurate to do measurements at several locations in the ield and obtain the values or α x and α y with a least squares it.

10 66 L. Noethe and S. Guisard: Field astigmatism in decentered telescopes M φ P cp y M β α z d Fig. 4. Change o α due to a rotation o M around its vertex We use typically eight measurements at evenly distributed points at the edge o the ield. Such eight measurements would at least take iteen minutes. During this time the VLT optics changes, because o elastic deormations due to changes o the zenith distance, signiicantly. In particular, the aberrations decentering coma and astigmatism are strongly aected. Thereore, we had to do a closed loop active optics correction at the center beore each measurement at the edge and, in addition, had to subtract the variation generated by the change o altitude between the correction at the center and the measurement at the edge. 4.. Measurement o the accuracy o the method By changing the misalignment in a well deined way it is possible to estimate the accuracy o this method. The VLT has got the useul eature that the primary mirror can be moved by motors in ive degrees o reedom. The only ull body movement which cannot be remotely controlled, is a movement in the direction which is perpendicular to the optical and to the altitude axis. This allows that, in particular, the primary mirror can be tilted arbitrarily around its vertex. We can thereore modiy the misalignment angle α between the primary and secondary mirror very accurately. By measuring aterwards α we can thereore check the accuracy o the method described above and, in addition, the validity o the theoretical parameters B 0, B and B. The expected change α o the angle between the axes o the primary and secondary mirrors due to a rotation o the primary mirror around its vertex can be deduced rom Fig. 4. In an initially perectly aligned telescope irst the primary mirror has been rotated around its vertex by φ. Aterwards decentering coma has been corrected by rotating the secondary mirror by β around its center o curvature. The axes o the primary and secondary mirrors then intersect at the coma-ree point P cp. For small angles we get β = d z + z φ. 83) The angle between the axes o the primary and secondary mirrors is then α = β φ. 84) The derivation o the change o the angle α between the axes o the primary and secondary mirrors does, at least or small initial misalignments, not depend on the actual initial state o the telescope. Equation 84) is thereore always correct. With the VLT parameters one gets α = 6.64 φ. 85) 4.3. Measurement data The irst mapping was done or the initial setup o the telescope. The second and third mappings were done ater tilting the primary mirror around it vertex around two orthogonal axes A and B each time by 0. For a rotation o the primary mirror around its vertex by 0 one expects a change α = 3.8.

11 L. Noethe and S. Guisard: Field astigmatism in decentered telescopes 67 In addition, a rotation around the vertex will shit the whole pattern due to the tilt o the M axis, but the shit is only o the order o 0.5 on the sky and thereore negligible. The three mappings gave the ollowing results or the x- and y-components o α all igures in arcseconds). φ α x α y α x α y α A B The total change α o the misalignment is deined by α = α x ) + α y ). The average o the measured changes o the misalignment angles α between the irst and the second coniguration on the one hand and the irst and the third coniguration on the other hand is α = The dierence to the expected value α = = 3.8 is only 5.3. I, ater a rotation o the secondary mirror around the coma ree point a similar accuracy is achieved, the two nodes o the binodal ield will only be 7 apart. At the edge o the ield o 0.5 this would lead to an error in the coeicient o third order astigmatism o 56 nm, which is negligible. 5. Summary Explicit expressions in terms o undamental telescope parameters o a two mirror telescope can be derived or the coeicients describing the ield dependence o third order astigmatism. O particular importance is the dependence o the coeicients on the position o the stop. In addition, these parameters are recalculated or a reerence system where the center o the adapter is at the origin. The numerical values are used to show that rom the measurement o the ield dependence o third order astigmatism at eight locations in the pupil the axes o the primary and secondary mirrors o the VLT can be aligned with an accuracy o approximately 7. Acknowledgements. The authors would like to thank Ray Wilson or helpul comments and discussions. Reerences McLeod B.A., 996, PASP 08, 7 Schroeder D.J., 987, Astronomical Optics. Academic, San Diego Shack R.V., Thompson K., 980, SPIE, Vol. 5, Optical Alignment, p. 46 Wilson R.N., 996, Relecting Telescope Optics I. Springer-Verlag, Berlin Wilson R.N., 999, Relecting Telescope Optics II. Springer-Verlag, Berlin