Requirements for Subreflector and Feed Positioning for ALMA Antennas

Transcription

1 ALMA Memo. No. 479 Requiements o Subelecto and Feed Positioning o ALMA Antennas Byan J. Butle National Radio Astonomy Obsevatoy 3-Dec-8 Abstact This memo pesents some calculations elated to equiements on subelecto and eed positioning o the ALMA antennas. In ode to keep gain loss at less than 1%, the axial (vetical positioning eo in the subelecto should be less than about.9 λ, and in the eed should be less than about.9 λ. The lateal positioning eo o the subelecto should be less than about.45 λ, and o the eed should be less than about 1 λ. The otational (tilt eo o the subelecto should be less than about 5.5 λ mm acminutes i the otation is about the vetex o the subelecto. I the otation is about the pime ocus, then this eo can be much lage: about λ mm degees. 1 Intoduction Incoect positioning o the subelecto and eeds o the ALMA antennas will esult in a loss in gain. Thee ae othe moe subtle eects (like changing the pimay beam shape and the phase acoss the pimay beam, which ae issues o mosaicing, but these ae not teated in this memo - only the loss in gain om eos in subelecto and eed positioning. Theoy This study is based nealy completely on an unpublished note witten by John Ruze in 1969 titled Small Displacements in Paabolic Relectos. A simila teatment o this poblem, but o displacements o the eeds o the MMA antennas, is in Shillue (1997. An unpublished note by James Lamb ( Veiication o Ruze Fomulas By Compaison with Ray-Tacing, dated 1-May-9 gives a coection o the omulas in Ruze s note, making sue that the sign conventions ae always consistent, and not nomalizing the path eo to the on-axis eo (this allows o the eects o multiple osets to be added in the apetue plane all at once. This note is included hee as Appendix A. Using pola coodinates in the apetue plane and φ, the phase eos esulting om incoect eed o subelecto positioning ae denoted δ(, φ. Given the subelecto osets, the om o δ(, φ can be explicitly witten, and o a Cassegain electo (as o ALMA ae shown in Table 1 (taken om Lamb 1. The geometical quantities in Table 1 ae: θ p = angle between optical axis and a ay om the eed to the subelecto; θ = angle between optical axis and a ay om the subelecto to the main electo; M = antenna magniication; c a = distance om pimay ocus to subelecto suace along the optical axis. 1

2 Table 1: Fom o Phase Eos o Given Subelecto Osets. type δ(, φ eed axial displacement ( z π z λ cos θ eed lateal displacement ( π λ sin θ p cos φ subelecto axial displacement ( z s π z s λ cos θp +cosθ subelecto lateal displacement ( s π s λ sin θp sin θ Λ cos φ π α c a subelecto otation ( α λ sin θp + M sin θ Λ cos φ Fo the ALMA antennas, M is, and to ist ode, we can assume that c a = /M o ocal length, which is 4.8 m o the ALMA antennas (Lamb Given the phase eo acoss the apetue, the loss in gain is calculated via (equation 3 o Ruze 1969: whee and δ = δ = G G o =1 δ + δ, (1 π 1 π 1 π 1 (, φ δ (, φ ddφ π 1 (, φ ddφ (, φ δ(, φ ddφ (, φ ddφ, (. (3 The unction (, φ is the eed illumination unction, which to ist ode can be assumed to be: ( = 1 a. Fo a 1dB Gaussian tape on the eed illumination, a =.75, which is the value used in this study. Beoe calculating the above integals, a best-it linea phase plane acoss the apetue must be subtacted (this is the equivalent o what oset pointing does o you. This is done by calculating the phase acoss the apetue and doing a least squaes it o the paametes o the plane, and then subtacting that best-it plane om the aw calculated phases beoe calculating the integals. Note that the situation o the ALMA antennas is moe complicated than this, because the eeds ae not all exactly on-axis. To account o the displacement om the optical axis, the subelecto will be shited lateally to eocus the beam onto the slightly o-axis eeds (Lamb et al. 1. The analysis hee is still coect to ist ode, though, and allows o limits to be placed on the deviation o the subelecto o eed om its nominal position, whethe that is on-axis ocus, o slightly oset to ocus o-axis. 3 Results The esultant cuves o loss o gain ae shown in Figues 1 though 5, as unctions o the positioning eos. In ode to have the loss in gain be less than 1% om the axial positioning eos, the eo in the subelecto

3 should be less than about.9 λ, and in the eed should be less than about.9 λ. The lateal positioning eo o the subelecto should be less than about.45 λ, and o the eed should be less than about 1 λ. The otational (tilt eo o the subelecto should be less than about 5.5 λ mm acminutes. 3.1 A Compaison An unpublished note by James Lamb ( Seconday Mio Positional Toleances, dated 1999-Jan-9, contains omulae o the eective ms intoduced by subelecto misalignments. This note is included hee as Appendix B. One slight dieence in that wok and this one is that the assumed illumination was dieent. James assumed uniom illumination, while this wok uses an illumination unction which appoximates a Gaussian tape with an edge tape value o -1 db. So slightly dieent esults should be expected, but they should be the same to ist ode. In any case, in ode to compae the esults hee with those, it is necessay to ist deive the elationship between eective ms due to optics misalignment and gain loss. Deine the apetue eiciency as: η a = η o e (4πσ /λ, (4 o wavelength λ and total ms σ. Assume that the total ms is made up o two tems: a main dish component (σ and an optics misalignment component (ɛ and that they add in quadatue o the total: σ = σ + ɛ. (5 Then the atio o the apetue eiciency with no optics misalignment to that with the optics misalinment is: R = e (4πɛ/λ. (6 Since ɛ is small, expand the exponential and dop all highe ode tems: ( 4 πɛ R =1. (7 λ Now, assuming that R =.99, as in the teatment above, leaves: Subelecto Lateal Displacement ɛ.8 λ. (8 James unpublished note gives: ɛ = a 1 x, (9 whee a 1 = 19.1 µm mm 1. Fo compaison with the esults o this memo, substitute x =.45λ, and ɛ =.8λ:.8λ = a 1.45 λ, (1 yielding a 1 = 17.8 µm mm 1. This is good ageement Subelecto Axial Displacement James unpublished note gives: ɛ = a z, (11 whee a = 8.7 µm mm 1. Fo compaison with the esults o this memo, substitute x =.9λ, and ɛ =.8λ:.8λ = a.9 λ, (1 yielding a = 88.9 µm mm 1. Again, petty good ageement. 3

4 3.1.3 Subelecto Rotation James unpublished note gives: ɛ = a 3 θ, (13 whee a 3 =8.16µm deg. Fo compaison with the esults o this memo, substitute θ =91.7λ (which is 5.5λ mm acmin, conveted to degees, and ɛ =.8λ:.8λ = a 3 (91.7 λ, (14 yielding a 3 =.951/λ µm deg. Hee, thee ae two immediately appaent poblems - the popotionality is dieent (the dependence on wavelength, and the magnitude is signiicantly dieent (ode o magnitude at λ = 1 mm: James esult would allow o a otational eo as lage as 1degat1mm,whiletheesult hee is that it can only be 6 acmin at that wavelength. Reducing James omula to the allowable eo as a unction o wavelength gives: θ λ mm degees. This dieence can be attibuted entiely to the assumption o whee the otational cente is. Ruze s omula assumes that the otation is about the vetex o the subelecto, while James assumes that the otation is about the pime ocus. These ae two endmembes in a sense - best and wost case scenaios. Until the design o the subelecto motion mechanism is inalized, it is diicult to say what the inal equiement will be on otation o the subelecto. At the time the design is inalized and the point about which eective otations occu is known, a new study should be completed and this issue can be settled. It should be noted, howeve, that otations about any othe point than the pime ocus can be decomposed into a lateal tanslation and a otation about the pime ocus. Fo this eason, i the subelecto measuement and adjustment scheme ist ixes the lateal tanslation and then the subelecto otation, the eective otation eo will be about the pime ocus. Futhemoe, otations about non-pime ocus locations cause waveont abeations which do not add in quadatue with the othe abeations. These two easons ae why James took the appoach that he did, and because o this his numbes will likely tun out to be close to the tuth. Acknowledgements Discussions with Pete Napie helped immensely. Obviously James Lamb s documents (Appendices A and B also helped consideably, as did discussions with him. Reeences Lamb, J.W., A. Bayshev, M.C. Cate, L.R. D Addaio, B.N. Ellison, W. Gamme, B. Lazae, Y. Sekimoto, & C.Y. Tham, ALMA Receive Optics Design, MMA Memo 36, 1 Lamb, J.W., Optimized Optical Layout o MMA 1-m Antennas, MMA Memo 46, 1999 Shillue, B., Gain Degadation in a Symmetical Cassegain Antenna Due to Lateally Oset Feeds, MMA Memo 175,

5 Figue 1: Loss o gain as a unction o eed axial oset, in wavelengths. Figue : Loss o gain as a unction o eed lateal oset, in wavelengths. Figue 3: Loss o gain as a unction o subelecto axial oset, in wavelengths. Figue 4: Loss o gain as a unction o subelecto lateal oset, in wavelengths. Figue 5: Loss o gain as a unction o subelecto otation eo, in acminutes, o an obseving equency o 65 GHz (λ 46 µm. 5

6 Appendix A 6

7 Veiication o Ruze Fomulas By Compaison with Ray-Tacing James W Lamb OVRO, Caltech Intoduction In many instances it is expedient to evaluate the eective path eo due to displacements o the eed o seconday mio in closed om. Ruze [1] has deived some simple omulas o this pupose. When these ae to be used with othe calculations, such as inite-element-analysis (FEA it is impotant that the coodinate systems and sign conventions ae clea and consistent. To veiy the om and sign o these omulas we compae them with esults om ay tacing in Mathcad. Coections ae made to the signs o the oiginal omulas, and it is pointed out that the nomalization to the path eo at the cente o the apetue can lead to conusion and incoect esults i cae is not taken.. Geomety The geomety is deined in Figue 1. Two coodinate systems will be used, a cylindical one with the z- axis along the boesight diection, and a ectangula system which ollows the ALMA convention o the antennas []. The cylindical system esults in simple equations, but the ectangula system may be easie to implement in an FEA analysis. Fomulas in both systems ae povided. The antenna has a diamete D, a ocal length, and a magniication M. y x z Figue 1. Geomety and coodinate system. Thid angle pojection is used, so the uppe view is om above the antenna pointed at zenith, and the lowe view is om the ea o the antenna. The antenna otates about the negative x axis down towads the hoizon position. 7

8 Veiication o Ruze Fomulas 1-May-3 Page 3. Cylindical Coodinates The simplest oms o the equations ae given in tems o the cylindical coodinates (,, z. Table I pesents the eo in this coodinate system. The ollowing points should be noted: The path eo is the actual path length minus the ideal path length. That is, i the omula gives a positive numbe, the path length om the antenna ocus to the antenna apetue plane is inceased. (The oiginal Ruze pape was inconsistent in its sign convention. Axial displacements ae in the positive z diection. Radial displacements ae along a vecto at an angle to the x axis. Tilts o the seconday mio ae about a vecto paallel to z and passing though the seconday mio vetex. Rotations about othe points ae a combination o this plus a lateal shit. The omulas ae oten enomalized by setting the path eo to zeo at the cente o the apetue. Table I shows both the nomalized and un-nomalized equations, whee these die. The unnomalized ones can be diectly compaed with the ay-tacing esults. Table I. Path length eo in tems o cylindical coodinates. Position Eo Actual path eo Nomalized to zeo on axis Feed axial displacement by z z cos( z cos( Feed lateal displacement by sin( cos( at angle Seconday mio axial 1 zs cos( p cos( zs cos( p 1 cos( 1 displacement by z s Seconday mio lateal s sin( p sin( cos( displacement by s at angle Seconday mio tilt aound vetex Whee the angles p and ae deined by sin( p sin( 1 M 1 M c a M sin( cos( sin( p The above equations wee compaed diectly with a Mathcad ay-tacing analysis which ensues that the sign convention is consistent and the magnitudes o the esults ae coect. The actual expessions implemented in the Mathcad document ae (with F = M: p atan atan F p 1 z cos p sin p 3 z s cos p cos p 4 s sin p sin p 5 y ( c a sin p Msin p p 1 p p 3 p 4 p 5 The ollowing gaphs show the compaison with the ay-tacing esults. 8

9 Veiication o Ruze Fomulas 1-May-3 Page 3 z 1mm 998 Cut Though Plane o Symmety mm z s mm 9985 s mm y deg Path eo [um] Position in Apetue [m] Ray tacing Ruze Figue. Axial movement o eed. z mm 1 Cut Though Plane o Symmety 1mm z s mm 5 s mm y deg Path eo [um] Position in Apetue [m] Ray tacing Ruze Figue 3. Lateal eed shit. 9

10 Veiication o Ruze Fomulas 1-May-3 Page 4 z mm 1 4 Cut Though Plane o Symmety mm z s 1mm s mm y deg Path eo [um] Position in Apetue [m] Ray tacing Ruze Figue 4. Axial movement o seconday mio. z mm Cut Though Plane o Symmety mm z s mm 5 s 1mm y deg Path eo [um] Position in Apetue [m] Ray tacing Ruze Figue 5. Lateal movement o seconday mio. 1

11 Veiication o Ruze Fomulas 1-May-3 Page 5 z mm mm Cut Though Plane o Symmety z s mm s mm y 1deg Path eo [um] Position in Apetue [m] Ray tacing Ruze Figue 6. Tilt o seconday mio. As seen in Figue Figue 6 the equations do agee well with the numeical esults, and in paticula the signs ae consistent. The calculations also veiy that the path eos may be simply added to give the same esults as ay tacing. 4. Catesian Coodinates In Catesian coodinates the omulas take the oms shown in Table II. To compae with the pevious section, the lateal displacements and tilts ae decomposed into two components paallel to the x and y axes: x cos( y sin( xs s cos( xs s sin( x s sin( cos( y s Note that the otations x and y ae taken as positive otations aound the x and y axes espectively. The axis o otation passes though the vetex o the seconday mio. 11

12 Veiication o Ruze Fomulas 1-May-3 Page 6 Table II. Path length eo in tems o cylindical coodinates. Position Eo Actual path eo Nomalized to zeo on axis Feed axial displacement by z cos( z Feed lateal displacements Seconday mio axial z cos( 1 x cos( y sin( sin( by x, y zs cos( p cos( zs cos( p 1 cos( 1 displacement by z s Seconday mio lateal xs cos( y s sin( sin( p sin( displacements by x s, y s Seconday mio tilts by Whee x sin( y cos( c asin( p M sin( x, y sin( 1 M 1 M p sin( x y tan 1 ( y / x 5. Discussion The omulas pesented by Ruze [1] have been shown to be accuate o small displacements o the seconday ocus o seconday mio in the ALMA antennas once the coect signs ae used. It is citical that when the esults ae combined with FEA the signs must be consistent since the eects may be additive. An example is whee the pimay mio opens out unde gavity and pulls the seconday mio in close to the pimay. The ocus o the pimay moves out om the antenna, but the seconday moves in opposite diection exacebating the eect. Anothe pitall is that the omulas ae usually e-nomalized to ee them to the path length at the cente o the antenna apetue (This aects only the expessions o displacements along the antenna axis. This can be conusing since the sign o the actual path change given by z cos( has the opposite sign om the nomalized vesion z 1 cos(, though the shape o the unction is the same. I the equations ae to be applied to pat o the pimay mio (to ind the eect o panel displacements, o example then the un-nomalized omulas must be used. Anothe case whee the distinction is impotant is in the estimation o the phase eos induced. We would ecommend, theeoe, that the un-nomalized oms should always be used. This will not aect the calculation o the contibution to the waveont o eective suace eo. Reeences [1] J. Ruze, Small displacements in paabolic electos, Unpublished, Feb [] H. Riewaldt, Basic antenna deinitions, NRAO, ALMA-US ICD No. 11, Jan.. 1

13 Appendix B 13

14 9 JANUARY, Seconday Mio Positional Toleances James W Lamb Abstact Toleances o the positioning o the seconday mio ae deived om the associated abeations. The optimum location o the seconday will be detemined by adiometic measuements at egula intevals. It is the pecision o these measuements that should limit the accuacy, athe than the mechanics o the tanslation stage. The tanslation stage o the seconday should have a esolution o ~5 1 µm, and a lineaity o. % on each axis. I. INTRODUCTION The seconday mio will have a ocus-tanslation stage allowing ocus optimization duing opeation. Requiements on the positioning accuacy ae set by the contibutions allocated to the suace accuacy, pointing and phase eo budgets. To undestand how to speciy these it is necessay to know exactly how they ente into these budgets. Since the seconday mio has cicula symmety thee ae ive paamete which need to be speciied, thee tanslational and two otational. The tanslation eos ae x and y othogonal to the optical axis, and z paallel to it. Tilt eos ae denoted by θ x and θ y aound the x-axis and y-axis espectively. To detemine the eect o these eos ay tacing was used to calculate the waveont eo. This is slightly moe accuate than some o the analytical omulas which ae valid only o lage ocal atios. Results quoted hee assume uniom apetue illumination, but including a 1-dB tape does not change the esults signiicantly. The antenna paametes used in the calculations ae given in Table I The esults will be given in tems o an eective suace eo (hal waveont eo which would be added in quadatue to othe components o the suace accuacy budget. TABLE I: ANTENNA PARAMETERS USED IN THE TEXT Paamete Symbol Value Pimay diamete D 1. m Pimay ocal length 4.8 m Seconday diamete d 75 mm Magniication M Januay 9, 1999 Owens Valley Radio Obsevatoy, Calionia Institute o Technology, Big Pine, CA 93513, USA II. SETTING PROCEDURE The antenna pimay mio will be set up initially using a theodolite and tape. This should achieve a suace accuacy o ~1 µm, and eeence the optical axis to the gavitational vetical. The suace will then be measued using a hologaphy eceive mounted at the pime ocus and adjusted accoding to the suace eos deived om this. This eectively deines the location o the pime ocus elative to the hologaphy eed phase cente. Once the seconday mio has been installed it will be aligned so that it is pependicula to the optical axis. Since thee is no povision o any tilting mechanism, this will be done with shims, manual adjustment scews, o by machining. The axial and lateal (adial ocus positions will then be detemined om astonomical obsevations. It will be assumed o this analysis that these measuements ae made at a wavelength λ = 1 mm. At shote wavelengths the adiometic sensitivity will be lowe, but the equied pecision as a action o the wavelength will be less. The wavelength is theeoe not citical. Pointing and ocus will be checked evey 3 min and, i equied, the ocus optimized to the accuacy o these measuements. The tanslation stage needs to be accuate enough to tack ocus changes between these measuements. Peiodic calibations o phase will be made at intevals o 3 min on souces oset on the sky by up to. I the ocus needs to be changed between these calibations, the z- tanslation accuacy needs to be within the pathlength eo budget. III. GRAVITATIONAL CHANGES As the antenna moves in elevation the position o the seconday will need to be adjusted to compensate o gavitational changes in the pimay and suppot legs. The total movements, x tot and z tot will be on the ode o a millimete. Both o these will be sinusoidal unctions o elevation so that the maximum ate o change will be.17 x tot and.17 z tot pe degee o elevation change. Ove a 3-min inteval between phase calibations the elevation will change by at most.75 and the ocus will change by 13 µm o x tot o z tot o 1 mm. The aveage change is hal that. Fo changes between the souce and the phase calibato needing a elevation change, the ocus change will be ~35 µm. Ove the 3 min between pointing 14

15 LAMB: SECONDARY MIRROR POSITIONAL TOLERANCES and ocus deteminations the changes will be about ten times lage. IV. LATERAL ERRORS Eos in the lateal positioning o the seconday, x o y poduce coma and pointing osets. The path eo (and theeoe the contibution to eective suace eo inceases linealy with the oset. The contibution to the suace eo is ε e = a1 x (1 1 a1 = 19.1 P mm and the associated pointing change is x θ sky. 75 ( o about µm o a thitieth o a beam at λ 1-mm. Simila expessions apply to y. To adjust the lateal ocus a gain cuve can be made. This equies measuing the signal om a point souce o dieent lateal positions o the seconday with the pointing oset coected using (. An altenative way is to measue the asymmety o the ist sidelobes which esults om the coma. An imbalance o 1.3 db esults om an oset o x =.1 mm at λ = 1 mm, and this is associated with ε e = 1.9 µm. V. AXIAL ERRORS Axial eos, z, poduce spheical abeation. The pathlength and suace eo contibutions ae again in linea popotion to the displacement. Fo the nominal antenna paametes the contibution to the suace eo is ε e a = a z = 8.7 P mm To detemine the optimum ocal setting, the seconday would be scanned in z and the signal om an astonomical point souce measued. The hal-powe points ae at about a halwavelength on eithe side o the ocus. We can estimate that i we measue the ocus cuve at z ±λ/, a elative accuacy o % between the two points gives a measuement eo o the ocus position o ~. λ. At 1-mm wavelength the eo is theeoe about µm and (3 then gives ε e = 1.6 µm. Using the whole gain cuve would give a bette detemination o the ocus. 1 VI. ORTHOGONALITY OF AXES I the x- o y-axis is not pependicula to the optical axis, o the z-axis is not paallel to it, the lateal and axial eos will be intemixed. The wost case is an x- o y-tanslation having a component along the optical axis. Since the z-ocus is ive (3 times moe sensitive than the x o y it would take an angula oset o ~1 in the x- o y-axis to have an eect at the 1 % level. The main eect would be a change in the shape o the gain cuve (o measued beam shape. Aound the nominal ocus the symmety o the measuement is not aected so the alignment pecision is not educed. Even o elatively lage misalignments, only a ew iteations o the x, y, and z-stages would be equied to convege. VII. TILT ERRORS The minimum abeations occu o tilts aound an axis though the pime ocus. Rotations aound any othe axis can be decomposed into a otation aound the pime ocus and a tanslation. The tanslation pat is coveed in section IV. A tilt o θ poduces astigmatism with a contibution to the eective suace accuacy o ε e = a3 θ a = 8.16 P deg 3 The eeence axis deining zeo tilt is the optical axis om the eed to the pime ocus. This does not have to be coincident with the axis o the pimay mio paaboloid. It can be detemined using a theodolite and level to look diectly up at the pime ocus (deined, say, by the hologaphy eed om the nominal eceive location. Finding the pime ocus to within 1 mm deines the optical axis to bette than.1 elative to gavity. Leveling the seconday mio to this pecision then gives a maximum eo o. so that the contibution to suace eo is less than.3 µm. Note that a hologaphic measuement om the seconday ocus would measue astigmatism due to the seconday tilt. It could then be emoved by panel adjustment deived om the coesponding hologaphy map. VIII. ADJUSTMENT RANGE To get a good ocus detemination a ange o ± λ is equied on all axes. At 3 GHz this means a total tavel o mm. Any allowances needed o manuactuing and assembly toleances should be added to this. IX. DISCUSSION All the eos due to ocus setting as calculated above ae added in quadatue giving an RSS (oot sum o squaes o ε e = 3.6 µm. To achieve this the contibutions o the tanslation stage encodes and dives must be negligible. The esolution and epeatability o the encodes should be a ew times bette than the equied accuacy shown in Table II, o 5 1 µm. - (4 15

16 9 JANUARY, TABLE II: REQUIRED SETTING ACCURACY OF SECONDARY MIRROR AND THE ASSOCIATED CONTRIBUTION TO THE SURFACE ERROR BUDGET Paamete Accuacy Suace Eo x 5 µm 1.9 µm y 5 µm 1.9 µm z µm 1.6 µm θ x..3 µm θ y..3 µm Total (RSS 3. µm Focus measuements will be made ove a ange o about ±λ. At 3-mm wavelength this will impose a lineaity equiement o about. % within this ange. I the z-ocus is tacked duing an obsevation it will also aect the phase. In ode to know the pathlength to bette than 5 µm the encode would have to have an accuacy o.5 µm o bette ove the tacking ange. Section III showed that the ocus change ove 3 min on the souce is small enough to be ignoed. The ocus change going to the phase calibato may be lage, but since the calibation is at a longe wavelength this ocus adjustment can be omitted. The intevals between ocus and pointing calibations occu on the longe inteval o 3 min. Fo the maximum elevation change o 7.5 duing this time, the ocus could change by up to 13 µm. The seconday would need to be adjusted duing this peiod at each phase calibation with a pecision o ~1 µm. This also ensues that the contibution to the pointing eo is less than.3 acsec peak (<.15 acsec RMS The abeations ae unctions o the ocal length o the pimay. Lateal abeations vay as -, axial ones as ~ -, and tilts ae popotional to. Although the toleances become smalle o shote ocal lengths, the ocus cuves ae shape and the elative estimation o the optimum ocus emains about the same, independent o None o the measuements above can account o changes in the seconday position due to themal changes o wind oces which occu between ocus checks. A quadant detecto could be used to mitigate these vaiations. Existing systems meet the equiement o measuing to 1 µm o bette. The basic equiements o the seconday mio mount ae summaized in Table III. Note that these ae maximum eos so the RMS eos will be about a acto o two less. TABLE III: RECOMMENDED PARAMETERS FOR SECONDARY MIRROR FOCUS- TRANSLATION STAGE Paamete Value Resolution in x, y 1 µm Resolution in z 5 µm Lineaity in x, y. % Lineaity in z. % Range in x, y mm Range in z mm Tilt o mio elative to optical axis. Othogonallity o axes 1 16