Generalzed atmospherc dsperson correctors for the Thrty Meter Telescope Bran M. Sutn, TMT Observatory Corporaton ABSTRACT The Thrty Meter Telescope (TMT) s unbaffled and has stablty requrements tghter than the prevous generaton of 10- m class telescopes, leadng to tougher requrements on atmospherc dsperson correctors (ADCs). Snce nstruments are nternally baffled, ADCs may no longer shft the poston of the telescope ext pupl. Desgns that control pupl poston are explored. Keywords: atmospherc dsperson correctors, astronomcal telescopes, telescopes 1 INTRODUCTION When lght enters the atmosphere at an angle, the ar dsperses the color slghtly n angle. Ths s an ssue for spectrometers, snce the lght wll not all go nto a narrow slt. An atmospherc dsperson corrector (ADC) s used to correct ths. Usually an ADC conssts of several wedges of glass, sometmes of dfferent types of glass, whch translate or rotate to actvely correct the atmospherc dsperson, as a functon of the elevaton angle of the telescope. Another ssue s anamorphc dstorton, whch s mportant for a wde-feld nstrument. The atmospherc refracton compresses the feld n the zenth drecton. For magng, ths effect can be corrected n software, but for spectroscopy wth a wde-feld slt mask, the objects move relatve to the slts as the feld rotates. The ADC can also be used to correct ths effect. On the telescope sde, TMT s beng bult wthout baffles n order to decrease wnd loadng, and all nstruments are expected to nternally baffle for stray lght. Ths means that an nstrument must frst have a cleanly formed nternal pupl, and the pupl of the telescope must be statonary relatve to the nternal bafflng. The second ssue s that the TMT tertary mrror can tp and tlt, but not actvely translate. Gravtatonal deformaton of the structure leads to translaton of the tertary, whch translates both the pupl and the mage. Ths paper addresses both, snce some ADC desgns do not mantan a statonary pupl. 2 ATMOSPHERIC REFRACTION AND DISPERSION To frst order, the atmosphere s a bg plate of ar above the telescope. The lght enterng the atmosphere at an angle s refracted and dspersed just lke a prsm (Fgure 1). For a sngle-slt spectrometer, the dsperson has the effect that the lght s spread out as a functon of wavelength and may not all pass through the slt, dependng on the angle of the slt to the horzon. A second, more obscure effect s that the atmospherc refracton causes anamorphc dstorton of the sky mage, so the plate scale s dfferent dependng on the angle to the horzon. For long observatons usng wde-feld magng spectrometers, the effect can also cause lght to mss the slt as a functon of the horzon angle. Fgure 1: atmospherc dsperson bends short wavelengths more
For typcal temperature, pressure, and humdty of TMT located at Mauna Kea, the devaton of a sky object at 65 zenth angle s 77 arcseconds at a wavelength of 0.5 mcrons. The dsperson s 7.3 arcseconds/mcron, and the anamorphc dstorton s 1.001 1,3. For the 20-arcmnute feld of vew of TMT, ths anamorphc dstorton leads to a relatve mage moton of 1.2 arcseconds, whch s sgnfcant when compared to the typcal 0.75-arcsecond slt used for spectrometry. 3 TELESCOPE ABERRATIONS TMT s a classcal Rchey-Créten Cassegran telescope wth a flat tertary mrror that repostons the telescope focal surface to nstruments on the Nasmyth platform. Snce the telescope has no bafflng to keep stray lght from the focal surface, nstruments are expected to nternally baffle for stray lght by havng an nternal stop conjugate to the telescope ext pupl. If the tertary mrror rotates about the ntersecton pont between the telescope axs and the elevaton bearng, then any moton of the tertary repostons the focus and the telescope ext pupl together. Fgure 2: Tertary translaton moves the mage, the pupl, and the feld curvature n the azmuth drecton Ths happy story breaks down n practce because TMT s a very large steel structure whch flexes under gravty dfferently for each telescope elevaton angle, so the tertary does not reman centered on the ntersecton of the two prevously mentoned axes. If the tertary s translated orthogonal to ts own surface (n pston), then the telescope focus and the apparent ext pupl poston both move at the nstrument focal surface (Fgure 2). For an nstrument located on the elevaton axs, ths moton s n azmuth. For other nstrument locatons, ths apparent moton at some angle whch depends on the nstrument locaton on the Nasmyth platform. In practce, the telescope control system wll automatcally react to keep the gude object on the guder. If the tertary s rotated to keep the pupl on axs, then result s that the telescope s operatng off-axs. Snce a Cassegran telescope has feld curvature, the dfference between the before and after focal surfaces s a defocus that s lnear across the entre feld of vew. For a wde-angle nstrument, ths s not acceptable. One possble way to correct ths s to rotate the secondary about the coma-free pont; however, ths ntroduces bnodal astgmatsm, whch s also an aberraton whch goes lnearly wth feld angle for small feld angles. Even f the bnodal astgmatsm s small, t wll be a functon of elevaton angle and therefore change throughout an observaton, makng calbraton dffcult. The telescope aberratons are all constant f the focus shft s corrected by the ADC, rather than by repontng the telescope. If the tertary s moved perpendcular to t surface by a dstance E, then the focus moves sdeways n the azmuth drecton by a dstance E/ 2. The pupl angle s shfted by E/( 2L), where L (=46.4m for TMT) s the dstance from the focus to the pupl. If the focal length s f (=450m for TMT), then the telescope re-ponts by an angle E/( 2f). Rotatng the tertary by an angle τ moves the pupl by 2τ and the focus by 2τB, where B (=20m for TMT) s the dstance from the focus to the tertary. If the curvature of the focal surface s C (=1/(3m) for TMT), then the lnear defocus across the feld s ½ Cx 2 ½ C(x-E/ 2) 2 = CEx/ 2 plus a constant offset, where x s the mage moton n meters.
4 TYPICAL ADC S ADCs are, by and large, made from wedges of glass that act as prsms. These wedges can be ether sngle materals or glued-up stacks of materals wth dfferng refractve propertes. Fgure 3 below shows one popular type of ADC, commonly called a Rsley prsm. The two wedges are close to each other and counter rotate. When the wedges are opposte, as n the left sde of Fgure 3, any dsperson created by the frst surface s canceled by the last surface. When the wedges are rotated about the optcal axs so that the thn edges are together, then dsperson (and devaton) s maxmzed. Fgure 3: Counter-rotatng ADC (Rsley prsm) The other common ADC desgn s the trombone ADC, where one wedge s fxed and the other wedge translates along the space between the fxed wedge and the focal surface. Ths s shown n Fgure Fgure 4 below. When the movng wedge s touchng the fxed wedge, the system has no dsperson or devaton. When the movng wedge s at the focal surface, t has vrtually no dspersve effect on the mages, and so the maxmum dsperson (due to the fxed wedge) occurs. Fgure 4: Trombone ADC Both of the above desgns have the desred varable dsperson, whle only havng one degree of freedom. Alternate desgns may be able to compensate for more atmospherc and telescope mechancal effects. 5 OPTICAL CONSIDERATIONS Takng a perfectly good telescope and stckng a few wedges of glass nto the beam has all sorts of sde effects besdes cancelng the atmospherc dsperson. The effects consdered here are: The telescope mage has anamorphc dstorton (cancelng the atmosphere). The telescope mage s dspersed (cancelng the atmosphere). The telescope mage s dsplaced (also caused by telescope gudng). The telescope ext pupl s dspersed. The telescope ext pupl s dsplaced (also caused by tertary gudng). Lnear defocus across the feld (also caused by telescope pontng). The effects not consdered here are:
3 rd order aberratons caused by addng optcal materal to the lght path Second-order effects (e, assume the effects of ndvdual wedges lnearly add) Stray lght, surface qualty, ndex nhomogenety, scatter, absorpton 6 BACKGROUND MATHEMATICS We begn my assumng a stack of wedges of glass, and that all of our angles are small enough that we can lnearze Snell's Law. Snce each astronomcal observaton has some unque wavelength passband, we correct the atmospherc dsperson at two wavelengths, λ 1 and λ 2. Defne three functons of the ndex of refracton of the wedge materal n(λ) as and The beam devaton P for the th wedge s then n= n(λ 1)+n(λ 2 ), (1 2 Δ n=n(λ 1 )+n(λ 2 ) 2, (2) δ n= n(λ 1) n(λ 2 ) λ 2 λ 1. (3) P =Δ n A. (4) where A s the wedge angle. Each wedge has two surfaces, but the factor of two has been absorbed nto the constructon of Δn. Here both P and A are 2-dmensonal vectors n a plane parallel to the focal surface, so as a functon of rotaton angle θ about the nstrument rotaton z-axs, A s A = A [ cos(θ ) sn(θ ) ] where the coordnates are [ The wedge also has dsperson, whch has a smlar formula, azmuth zenth angle]. (5) p =δn A. (6) The locaton of the telescope pupl, as seen from the nstrument, only depends on the beam devaton. If the pupl locaton s specfed as the angular moton of the center of the pupl relatve to the z-axs, as seen by the focal surface, then the pupl moton s the same as the beam devaton, and the pupl dsperson s the same as the beam dsperson. The locaton of the focus at the nstrument s shfted by each wedge n proporton to the dstance from the wedge to the focal surface z, so F = 1 f z Δ n A (7) and smlarly for the dsperson, d = 1 f z δn A. (8) The f here s the focal length of the telescope, whch has been added so that F and d have unts of angle on the sky. The lnear defocus from a wedge s the change n optcal path length through the wedge,
where the defocus at dstance x from the center would be H x. H =½ Δ n A (9) The anamorphc dstorton of a wedge depends on the tlt of the wedge about the axs of the wedge's dhedral angle. If the wedge tlt s symmetrc (as a mnmum-devaton prsm), then there s no anamorphc dstorton. Call ths tlt angle φ=0. On the other hand, f the wedge s rotated so one face s normal to the z-axs, then φ=½ A, whle the anamorphc dstorton s sec(a+p) 1 + ½ (na) 2. So the anamorphc dstorton from a wedge s S 1=n 2 ϕ A. (10) The anamorphc dstorton from two wedges multples, but once second-order terms are dscarded, ths s equvalent to addng up the rght-hand sdes of equaton (10). For an ADC made up of a set of wedges, the total ADC effect s then P= Δ n A p= δn A F= 1 f z Δ n A d= 1 f z δn A H=½ Δ n A S=1+ n 2 ϕ A (pupl shft) (pupl dsperson) (pontng shft) (dsperson) (lnear defocus) (scale change). (11) From the prevous sectons, the left-hand sdes of these equatons are (for TMT) desred to be P = [ 2 E /L+2 τ x+α x 2 τ y +α y ] p = [ 0 0] F = [ 2 E/f +2 τ x B /f +α x 2 τ y B/f +α y ] d = [ 0 d sky] H = [ 2Cf α x 2Cf α y] S = [ 1 S sky]. (12) Other telescopes such as GMTO or E-ELT would have dfferent requrements. Here τ x,τ y are the azmuth and zenthangle tlts of the tertary, whle α x,α y are the azmuth and zenth-angle motons of the telescope on the sky. Equatons (11) and (12) combne (completely gnorng sgns) to gve
2E / L+2 τ x +α x 2 E+2 τ x B+f α x = 0 = = 0 = 4 Cf α x = 0 = A cos (θ ) Δ n 2 τ y +α y = A sn(θ )Δ n A cos (θ )δ n 0 = A sn(θ )δn A cos (θ ) z Δ n 2 τ y B+f α y = A sn(θ ) z Δ n A cos (θ ) z δn f d atm = A sn(θ ) z δn A cos (θ ) Δ n 4 Cf α y = A sn(θ )Δ n A cos (θ )n 2 ϕ S atm 1 = A sn(θ )n 2 ϕ. (13) 7 SINGLE-MATERIAL CORRECTORS The TMT focal surface s about 2.6 meters n dameter. The only refractve materal that can reasonably be sourced n ths sze s fused slca. Under the assumpton that all the wedges are made from the same materal, equaton (11) smplfes to P /Δ n= p/δn= Ff /Δn= df /δn= 2H /Δ n= (S 1)/n 2 = A A z A z A A A ϕ. Note that P, p, and H have become degenerate, as well as F and d. The concluson s that an ADC made from a sngle materal cannot control the devaton and the dsperson separately, ether for the pupl or for the mage. Ths s regardless of the number of wedges n the ADC. The only algorthm avalable s then: 1) Set ADC angles and z-axs postons to correct atmospherc dsperson (d) wth no pupl dsperson (p) 2) Set telescope and tertary to correct mage offset (F) and pupl offset (P) 3) Set ADC wedge tlts to correct anamorphc dstorton (S) Ths s essentally the classcal ADC control algorthm, where the ADC corrects zenth-angle effects, whle the telescope corrects azmuth effects. The ADC s not able to offset lnear defocus from the telescope. Assumng only two wedges, equaton (13) smplfes to gve 8 TWO-WEDGE CORRECTORS (14) 2E /L+2 τ x +α x =Δ n 1 A 1 cos(θ 1 )+Δ n 2 A 2 cos(θ 2 ) 2 τ y +α y =Δ n 1 A 1 sn(θ 1 )+Δ n 2 A 2 sn(θ 2 ) 0=δn 1 A 1 cos (θ 1 )+δn 2 A 2 cos(θ 2 ) 0=δn 1 A 1 sn(θ 1 )+δn 2 A 2 sn(θ 2 ) 2 E +2 τ x B+α x =z 1 Δ n 1 A 1 cos(θ 1 )+z 2 Δ n 2 A 2 cos(θ 2 ) 2 τ y B+α y =z 1 Δ n 1 A 1 sn(θ 1 )+ z 2 Δ n 2 A 2 sn(θ 2 ) 0=z 1 δn 1 A 1 cos(θ 1 )+ z 2 δn 2 A 2 cos(θ 2 ) f d atm =z 1 δn 1 A 1 sn(θ 1 )+z 2 δn 2 A 2 sn(θ 2 ) 4 Cf α x =Δ n 1 A 1 cos(θ 1 )+Δ n 2 A 2 cos(θ 2 ) 4 Cf α y =Δ n 1 A 1 sn(θ 1 )+Δ n 2 A 2 sn(θ 2 ) 0=n 1 2 A 1 cos(θ 1 ) ϕ +n 2 2 A 2 cos(θ 2 )ϕ 2 S atm 1=n 1 2 A 1 sn(θ 1 ) ϕ 1 +n 2 2 A 2 sn(θ 2 )ϕ 2. (15) The two requrements for the pupl dsperson on the second lne result n two condtons, δ n 1 A 1 =δ n 2 A 2 θ 1 =θ 2 +180. (16)
The conclusons from ths are Requrng no pupl dsperson requres the trombone desgn; counter-rotatng wedges dsperse the pupl. Because δn depends on wavelength, the two wedges must be dentcal f the ADC s to work perfectly at arbtrary wavelength bandpasses. Usng (16) and assumng θ 1 = 90 to smplfy (15) gves 2 E /L+2 τ x =0 2 τ y +α y =Δ n 1 A 1 Δ n 2 A 2 ( P) 2 E+2 τ x B=0 2 τ y B+α y =z 1 Δ n 1 A 1 z 2 Δ n 2 A 2 (F ) f d atm =δn 1 A 1 ( z 1 z 2 ) ( D) α x =0 4Cf α y =Δ n 1 A 1 Δ n 2 A 2 (H ) S atm 1=n 12 A 1 ϕ 1 n 22 A 2 ϕ 2 (S) (17) The two-wedge ADC has no effect on the azmuth drecton at all. The control algorthm s then 1) set the dstance between the two wedges n lnear proporton to the mage dsperson (d) 2) set telescope α y to cancel the ADC lnear defocus (H). 3) set the tertary τ y to cancel the telescope and ADC pupl shft (P) 4) slde the entre ADC n z to cancel the focus shft (F) 5) set the wedge tlts to correct the anamorphc dstorton (S). So a two-wedge, mult-materal ADC can, n theory, solve all the problems n the zenth drecton, the performance s not really sgnfcantly better than a sngle-materal ADC, f at all. 9 MULTI-WEDGE CORRECTORS The process of usng snes and cosnes as n the two-wedge desgn clearly becomes a morass for mult-wedge ADCs. Instead, a graphcal approach makes more sense. Consder each term n the summatons as a vector n (x,y) or (azmuth,elevaton) space. The length of a vector term for, say, p, s δn A, and the drecton s θ. These vectors can be placed end-to-end on a 2-D (x,y) plot to represent the summatons. Snce p s a zero-length vector, the vectors created a closed, drected polygon. Thus for a two-wedge ADC, the vectors must be equal and opposte, exactly as n equaton (16). See the left-hand sde of Fgure 5 for what these graphs look lke. The short arrow n the d-plots represents the atmospherc mage dsperson to be matched by the ADC. For the three-wedge ADC, the three vectors for p form a trangle. Gven that the lengths of the vectors are fxed by the materals and shapes of the elements, the only modfcaton that can be made to the wedge postons whle keepng p zero s to rotate the trangle as a whole. Snce ths rotaton s reserved for algnng the dsperson wth the elevaton drecton, the wedges cannot ndependently rotate. Fgure 5: graphcal llustraton of ADC pupl and mage dsperson The concluson s that a three-wedge ADC s only margnally better than a two-wedge ADC, n that the extra z-moton of the extra wedge allows for correcton of focus shft wthout movng the entre ADC. Also, the pupl dsperson can most lkely be corrected over an arbtrary passband. By consderng drected graphs, t s clear that a 4-wedge ADC has sgnfcantly more flexblty compensate for varous ssues, but for a wde feld of vew, a four-wedge, mult-materal ADC s qute unrealstc.
10 CONCLUSIONS Ths paper consders generalzed ADCs on the TMT to see f the ADC can compensate not only for atmospherc dsperson, but also for the translaton of the tertary. The concluson s that no ADC wth less than 4 wedges wll compensate for the tertary, snce the tertary moton s orthogonal to the dsperson drecton. A trombone ADC made from a sngle materal (or dentcal combnatons of materals) can compensate the dsperson over any passband wthout dspersng the pupl. Usng multple materals or an extra wedge does not sgnfcantly mprove the ADC performance. 11 REFERENCES [1] Stone, Ronald C. An accurate Method for Computng Atmospherc Refracton, PASP, 108, 1051-1058 (1996). [2] Flppenko, A. V., The mportance of atmospherc dfferental refracton n spectrophotometry, PASP, 94, 715-721 (1982). [3] Sutn, B. M., Skewray, <https://gthub.com/skewray/skewray/blob/master/test/atmosphere.sky>, (2016). The TMT Project gratefully acknowledges the support of the TMT collaboratng nsttutons. They are the Assocaton of Canadan Unverstes for Research n Astronomy (ACURA), the Calforna Insttute of Technology, the Unversty of Calforna, the Natonal Astronomcal Observatory of Japan, the Natonal Astronomcal Observatores of Chna and ther consortum partners, and the Department of Scence and Technology of Inda and ther supported nsttutes. Ths work was supported as well by the Gordon and Betty Moore Foundaton, the Canada Foundaton for Innovaton, the Ontaro Mnstry of Research and Innovaton, the Natonal Research Councl of Canada, the Natural Scences and Engneerng Research Councl of Canada, the Brtsh Columba Knowledge Development Fund, the Assocaton of Unverstes for Research n Astronomy (AURA) and the U.S. Natonal Scence Foundaton.