1. Instrumentation Concepts Ground-based Optical Telescopes Keith Taylor (IAG/USP) ‏ Aug-Nov, 2008 Aug-Sep, 2008 IAG-USP (Keith Taylor)‏

2. Telescope Focal Plane Slit Spectrograph collimator Dispersing element camera detector Figure 3.1

3. Where is the re-imaged pupil? (= Image of Telescope formed by Collimator) ‏ Collimator Camera Det Figure 3.2a Pupil

4. Astro. Spectrograph (Schematic) ‏  D D A (D)‏  (  )‏  1 (  )‏ a (d)‏ F1F1 F1F1 F2F2 T’scope Coll Cam Det Figure 3.2 x ( )‏ lxlx lyly y

5. Astro. Spectrograph (Schematic) ‏  D D A (D)‏  (  )‏  1 (  )‏ a (d)‏ F1F1 F1F1 F2F2 T’scope Coll Cam Det Figure 3.2 x ( )‏ lxlx lyly y

6. Astro. Spectrograph (Schematic) ‏  D D A (D)‏  (  )‏  1 (  )‏ a (d)‏ F1F1 F1F1 F2F2 T’scope Coll Cam Det Figure 3.2 x ( )‏ lxlx lyly y A = Telescope collecting areaD = diameter  = solid angle subtended at telescope aperture  = angle a = beam area of collimatord = diameter  1 = acceptance solid angle at spectrograph  = angle F 1 = focal-ratio of telescopeF 2 = focal-ratio of spectrograph camera dp = pixel-size of detectorl x -by-l y = detector linear dimensions

7. The wavelength resolving power (R) of an astronomical spectrograph is given by: R = /d Entendue (information flux) through any optical system (eg: telescope to spectrograph) is conserved and given by: A  = a  1 (or: D.  = d.  ) ‏ Source with surface brightness,  (ergs.s -1.cm -2.sterad -1 ) then flux gathered by spectrograph is: . .A.  (ergs.s -1 ) ‏ where: A  = Entendue = “Information Throughput” Luminosité =  A  (= L) ‏ LR-product =  A  R (a general “figure of merit”) ‏ Pre-area detectors: Says nothing about pixelation of data

8. NB: A  implies single circular apertures, but … Area detectors (eg: CCDs) allows 1-D (y) of spatial information 1-D (x) of spectral ( ) information Now a given pixel-size (dp) is given by: dp = d.F 2.d  = D.F 2.d  While spatial and spectral multiplexes are given by: M x = l x /(2dp) ; M y = l y /(2dp) ‏ We can therefore re-define a figure of merit as:  A  R M x M y where: A  = Entendue  A  = Luminosité  A .R = the LR-product So:(LR) M x M y =  A  R M x M y Remember: d.d  =D.d  Nyquist sampling Now includes area detector advantage

9. Our figure of merit now becomes … (LR) M x M y = ..(  /4) 2.D. .(d  /d ).d. M x M y = ..(  /4) 2.D. .(dp/d ).(1/F 2 ). M x M y = .(  /4) 2.R.(l x.l y /4).(1/F 2 2 ) This figure of merit implies that there is no advantage to Large telescopes (D) or Large Spectrographs (d) ‏ But it is dependant on: Camera f-ratio (F 2 2 ) which should be minimized (ie: as fast as possible), and Detector format (l x.l y )which should be maximized Angular Dispersion Pixel Dispersion Bigger telescopes give smaller pixels on the sky Cram more light into a given pixels

10. So … need larger telescope to deliver finer spatial resolution Practical constraints: Input aperture:  x  1” (seeing limit) ‏ Pixel-size: dp ~20  m (fabrication constraint) ‏ Camera f-ratio: F 2  2 (refractive) and > 1 (Catadioptic) ‏ Constraints 1&2 (remembering) implies F 2.D ~ 8m … and even 8m requires f/1 (Schmidt) cameras and what do you do for ELTs? Conclusion: Large telescopes do not improve information gathering capacity but do give improvements in “Information Density” in units of ergs.s -1.cm -2.arcsec -2 Need to offer improvements in spatial resolution through the use of Adaptive Optics (AO) ‏ dp = D.F 2.d 

11. Astro. Spectrograph (Schematic) ‏  D D A (D)‏  (  )‏  1 (  )‏ a (d)‏ F1F1 F1F1 F2F2 T’scope Coll Cam Det Figure 3.2 x ( )‏ lxlx lyly y A = Telescope collecting areaD = diameter  = solid angle subtended at telescope aperture  = angle a = beam area of collimatord = diameter  1 = acceptance solid angle at spectrograph  = angle F 1 = focal-ratio of telescopeF 2 = focal-ratio of spectrograph camera dp = pixel-size of detectorl x -by-l y = detector linear dimensions

12. The Large Telescope Game Once D > 4m then either (or both) ‏ F 2 < 2 (not easy) ‏  < 1” (requires AO) ‏ For D ~8m and above, AO is essential Unless objects are spatially resolved (like faint galaxies) ‏ For spectroscopy (gratings or FPs) ‏ d  /d is intrinsic (ie: fixed for a given configuration) ‏ This means that D  d  (1/F 2 ) … double bind: The larger the telescope … The larger the spectrograph, and … The faster the camera Spectrograph cost  D n where n >>1

13. VIMOS on the ESO VLT

14. WFOS for TMT

15. Diffraction Gratings (Littrow - simplest realization) ‏ dd   d Figure 3.3 Where:  = Blaze angle  = groove spacing d = Grating depth For constructive interference: if d  = 0 : 2sin  = m /  Differentiating, gives: d  /d = m/( .cos  where m= order of interference

16. Spectral Resolving Power (R) ‏ Now from d  /d = m/( .cos  ) ‏ R = /d = m. / .cos .d  … and from d.d  = D.d  this gives R = 2d.tan  / .D = delay (Grating) / delay (Telescope)  D D A (D)‏  (  )‏  1 (  )‏ a (d)‏ F1F1 F1F1 F2F2 T’scope Coll Cam Det Figure 3.2 x ( )‏ lxlx lyly y

17. Astronomical Grating (Surface Ruled) ‏ Diffraction limited resolution is when: .D = or R D = 2d.tan  / = md/ .cos  But: d/ .cos  = N = # of = # of recomb. Beams and so … R D = m.N (a general interferometric result – cf: FPs and FTSs)‏ A standard astronomical grating

18. Reflection Gratings (used in low order) ‏ Path difference between interfering rays is given by: AB & A’B’ =  (sin  +sin  ) ‏ Constructive interference when path difference is an integer (m) # of waves: m =  (sin  +sin  ) ‏ where: m=order of interference Angular dispersion is given by differentiating wrt , hence d /d  = .cos(  )/m Linear dispersion is given by: d /dx = (d /d  ).(d  /dx) ‏ = .cos(  )/mf 2 (see next figure) ‏ Multiple interfering beams  Figure 3.4 Where: f 2.d  /x

19. Generic Reflection Grating Spectrograph (NB:  takes –ve value in diagram) ‏ f 2.d  = dx Figure 3.5

20. Practical considerations Slit-width, not diffraction, limited ie: R D = m.N R = 2d.tan  / .D = delay (Grating) / delay (Telescope) ‏ However, note from Figure: Beams to/from grating cannot (generally) collide: Requirement to extract beam from grating Non-Littrow Separate Collimator and Camera f-ratios ie: F 1  F 2 – not constrained to be equal

21. Now recall grating equation Constructive interference gives  (sin  +sin  ) = m The Blaze Wavelength ( 0 ) is defined when:  =  =  & m=1, hence: 0 = 2. .sin     Normal  = Blaze angle

22. Slit-limited Resolving Power As before: R D = /d = mW/  but, in practice (slit-width limited): R is determined by slit width, s’ at detector, where s  = s’  ’ or s’F 1 = sF 2, where F n = f n /D n

23. Derivations of R Now: d = (d /dx).s’ = ( .cos  /mf 2 ).s.(F 2 /F 1 ) ‏ = s.D 1. .cos  /mD 2. f 1 but, W = D 2 /cos , so d = s.  /mF 1 W and hence … R = /d = m F 1 W/s.  Given: s =  f T =  f 1 & f T = f 1 = f T /D T = f 1 /D 1 R = m W/   D T

24. Practical Example Consider following example: m =1 ;   1 = 1200/mm ;  = 0.5” ; = 500nm & D T = 8m ; D 1 = 100mm If grating tilt  = 20  then:  = arcsin(m /   sin  ) ~15 , and R = 1,560 (cf: R D = 124,800) ‏

25. 2 Grating Configurations From Collimator To Camera From Collimator To Camera Normal Blaze Axis Blaze Axis    NO SHADDOWING LOSSES (Blaze to Collimator)‏ (Ebert Config.)‏ SHADDOWING LOSSES (Blaze to Camera)‏ (Non-Ebert Config.)‏ Figure 3.6

26. 2 Grating Configurations From Collimator To Camera From Collimator To Camera Normal Blaze Axis Blaze Axis    NO SHADDOWING LOSSES (Blaze to Collimator)‏ (Ebert Config.)‏ SHADDOWING LOSSES (Blaze to Camera)‏ (Non-Ebert Config.)‏ Figure 3.6

27. Basic relationships For both configurations the Grating Equation can be re- written as: m /  = 2.sin[(  +  )/2].cos [(  -  )/2] But from Figure 3.6, |  -  | =  : … hence at the blaze wavelength ( B ),  +  = 2  Therefore: m B /  = 2.sin .cos(  /2) But 0 = 2. .sin  and so … B = 0.cos(  /2)

28. Ebert or non-Ebert? Differentiating the grating equation, the dispersion at blaze wavelength becomes: d  /d = (sin  + sin  )/( cos  ) ‏ = 2.sin .cos(  /2)/( B.cos  ) Therefore, angular dispersion increases with  greater for Ebert condition, but realized spectral resolution (dp/d ) is not necessarily greater Depends on Anamorphic factor, where: Anamorphism (A m )= cos  /cos 

29. Grating Anamorphism (Non-Littrow) ‏ EbertNon-Ebert Monochromatic Image Blaze to Collimator Blaze to Camera Figure 3.7

30. Grating View Figure 3.7a

31. Pros/Cons of Ebert condition Ebert (Blaze to Collimator) - advantages:  1 Maximizes coverage Dispersion is less than with non-Ebert condition, but projected slit width is narrower: Resolution is therefore maintained Ebert (Blaze to Collimator) - disadvantages: Slit width can become smaller than 2 pixels: Undersampled ; loose resolution Camera may not be able to support expanded beam

32. Blaze Effect (schematic) ‏ Blaze function represents the Energy distribution into various orders of the grating: Determines spectrograph efficiency Simple diffraction theory gives: I = [sin 2 N  ’/sin 2  ’].[sin 2  /  2 ] or I = [Interference F n ].[Blaze F n ] or: I = IF. BF where:  ’ = phase difference between facets  = phase difference across facets N = # of recombining beams    b Figure 3.8

33. Shape of Blaze Function Note:  ’ = (  / ).(sin  +sin  ) ‏ ie: the Grating Equation for  ’ = m  and  = (  b/ ).(sin  +sin  ) The Blaze Function (BF) is maximum when:  =  (or m=0) ‏ equivalent to specular reflection

34. Intensity distribution through grating orders /N  /  Figure 3.9

35. Free spectral range and Finesse (for a grating) ‏ In reference to previous Figure and equations: Separation between orders (m=0 & 1) = /  Finesse = N = # of recombining beams NB: From R D = m.N General case for interferometry (FPs, FTSs etc) ‏ FTS: m = ~10 5 or more ; N = 2 FP: m ~ 10 3 ; N ~ 30 Grating: m ~1 ; N ~10 5 or more For grating: m = 1, 2, 3 … For m=1: R D = /d = N

36. Blazed Grating In figure, maximum blaze efficiency at m=0, but Angular dispersion: d  /d = m/ .cos  = 0 Therefore, at 0 th order, light is undispersed Blaze function needs to be peaked at m=1… how? Blazed Grating Modify phase delay (  ) by tilting the facets (  ) ‏

37. Inclining the grating facets Where  = tilt of facet  = Camera/Collimator angle i & r = angles between incident and emergent rays wrt the facet normal (FN) ‏  &  are the angles of the incident and emergent rays wrt the Grating Normal (GN) ‏  Figure 3.10

38. Blaze peak condition  = ( .cos  / ).[sin(  -  ) + sin(  -  )] Note:  = 0 when 2  =  +  Hence, from the Grating Equation ‏ m 0 = 2 .sin(  ).cos(i) where: 0 is the blaze wavelength Note, also:  = (  cos  / ).(sin  - sin  ) ‏ and from previous Fig: i =  & r =  So Blaze peak condition (  =0) occurs when i = r ie: simple specular reflection

39. Blaze Wavelength ( B ) ‏ Also from Grating Equation m B = 2 .sin .cos(  /2) ‏ Ideally, at the  ~40% points: where  =  /2  = 2m B /(2m  1) ‏  = 2m B /(2m+1) ‏ ie: for m=1  = 2 B &  = (2/3) B Also:    = B /m … for large m Figure 3.11

40. Real grating performance Physical models of grating behaviour must include: Maxwell’s equations for a proper analysis Polarization effects Sum of all blaze energy = 1 (Whole lives have been spent modeling gratings!) ‏

41. Littrow Configuration (some fiber spectrographs) ‏ For Littrow (  =0), so: m B L = 2  sin  & R = 2D 1 tan  /  D T and … B = B L cos(  /2) ‏

42. Eucalyptus Spectrograph at OPD

43. Problems with non-Littrow Spectrographs Beam dilation: Non-zero camera-collimator angle (  ) gives variable beam dilation (anamorphism) so camera design needs to be compromised See Fig. 3.5&7 Pupil imagery: Grating represents last surface of diffraction so operates like its own pupil irrespective of where the optical pupil (image of the telescope formed by the Collimator) is located