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

2. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ The Fabry-Perot (French translation = Perot-Fabry) Fabry-Perot etalon works by constructive interference of light from multiple reflections between two exactly parallel surfaces Usually a gap between two precisely manufactured plates of some material: glass for an optical FP; sapphire for infra-red FP The gap is usually air (or vacuum) Interference Filters (IFs) have high refractive index materials. Inner surfaces are coated with a reflective surface.

3. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ Fabry-Perot Light enters etalon and is subjected to multiple reflections Transmission spectrum has numerous narrow peaks at wavelengths where path difference results in constructive interference need ‘blocking filters’ to use as narrow band filter Width and depth of peaks depends on reflectivity of etalon surfaces: finesse

4. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ FP notation  = refractive index of air a = amplitude of incident ray r = fraction of amplitude reflected light at air/glass interface r’ = fraction of amplitude of reflected light at glass/air interface t = fraction of amplitude transmitted light at air/glass interface t’= fraction of amplitude transmitted light at air/glass interface l = gap between reflective surfaces  = angle of incidence m = order of interference = wavelength of radiation; k = wave number (=2  c/ )  = phase difference between successive reflective rays

5. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ Stokes' treatment of reflection at a surface The principal of reversibility states that if you reverse the direction of all rays at a surface, the amplitudes will remain the same. Reversibility implies: att’ + arr = a ; art + ar’t = 0 Therefore: r’ =  r ; tt’ = (1  r 2 )

6. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ Reflectivity = r Transmissivity = t = (1  r)  att’ atr 2 t’ atr 4 t’ atr 6 t’ atr 8 t’ Summing the complex amplitude from each transmitted ray gives: A T e i  = att’ + att’r 2 e i  + att’r 4 e 2i  + att’r 6 e 3i  + … A T e i  = a(1  r 2 )(1 + r 2 e i  + r 4 e 2i  + r 6 e 3i  + … And summing the series gives: A T e i  = a(1  r 2 ) 1  r 2 e i  l By inspection, the reflective rays obey: A R e i  = a(r + (1  r 2 ){r’e i  + r’ 3 e 2i  + r’ 5 e 3i  + …

7. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ Convert from amplitudes to intensities To get the transmitted intensity ( I T ) multiply Ae i  by its complex conjugate: a 2 (1  r 2 ) 2 1  r 2 (e i  + e  i  ) + r 4 I T = I 0 (1  r 2 ) 2 1  2r 2 cos  + r 4 or: I T = Now: cos2  = 1  2sin 2 , so: 4r 2 (1  r 2 ) 2 sin 2 (  /2)] [1 + I0I0 IT =IT =

8. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ … from which we can derive the Airy Function The phase difference is given by: 2 .2  d.cos   = … and the reflected intensity R = r 2 So … 4R (1  R) 2 sin 2 ( 2  d.cos  ) 1 + I0I0 IT =IT = The FP Airy Function

9. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ FP in reflection and transmission From the Airy Function: Peaks occur when: m = 2  l.cos  and as R  1: the peaks become sharper 4R (1  R) 2 2  d  cos  sin 2 ( ) 1 + I0I0 IT =IT = I T is a maximum when sin(  /2) = 0 ie: when: 2  l.cos  = m Constructive interference This is also the condition for when: I R is a minimum Destructive interference NB: For zero reflection all beams after the first reflection are destructive of this first reflection. Complementary to …

10. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ Free Spectral Range (FSR =  ) & Finesse ( N ) Overlapping takes place when order m of wavelength 2 = 1 +  falls on top of order m+1 of wavelength 1. ie: (m+1) 1 = m 2 = m( 1 +  ) Thus the free spectral range (  ) = /m = 2 2  l.cos  Returning to the Airy Function: I T = I 0 /2 when: 2  l.cos  ( 0 +  /2) sin () = 2R2R 1  R … where  = FWHM of Airy Function Thus the finesse, N R =  R 1  R … this is known as the reflective finesse

11. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ More games with Finesse The Airy Function now becomes: 4N24N2 22 2  l  cos  sin 2 () 1 + I0I0 IT =IT = It will also be noted that: N R =  /  = . R / = R /m … where R = /  This reduces to the familiar expression: R = m N R the product of the (order * # of recombining beams) In turn this means that we can identify the finesse N with the # of recombining beams (or the e-fold decay of that quantity). In practice the profile of an etalon does not achieve the theoretical sharpness, it is degraded by other contributions to the finesse: Aperture Finesse N A = 2  /m  Defect Finesse N D = /2  L N E -2 = N R -2 + N A -2 + N D -2

12. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ Problems with absorption in the coatings The Airy Function can be re-written, remembering that T = (1  R): = I0I0 ITIT (1  R) 2 + 4R.sin 2 (  /2) T2T2 T2T2 (1  R) 2 = 1 + F.sin 2 (  /2) 1 However, we may have absorption losses of an amount A, so we must write: R + T + A = 1 and so Max. Intensity = [1 + (A/T)] 2 1 For the maximum transmitted intensity the important quantity is (A/T). Even though A may be very small, if T is also small (hence R large), (A/T) may become large and the T max will may be very low. Eg: Let R = 99.7% and A = 0.2%  T ~0.1%: Max. Intensity ~11% Let R = 99.7% and A = 0.29%  T ~0.01%: Max. Intensity ~0.11% What is happening physically is that while for each reflection there is very little loss, the reflectivity is so high that there are many effective reflections and the total loss becomes large.

13. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏   R ~0.8 A  0 Purity = Integral within  Integral outside  =  2 ( )  N tan 2N2N  tan -1 ~0.7 (for all N ) ( Contrast = Min Max1 + R 1  R )2)2 = For R ~0.9, N ~30, Contrast ~360 FPs are “dirty” OK for emission-lines (HII regions) (Spirals) (PNe) Not OK for absorption-lines (Stars) (E-galaxies)

14. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ FPs in practice Fabry-Perots are generally used in high order m to give high resolving power, the free spectral range (FSR = /m) is therefore small. Large gap (d) gives: Large order (m = 2  d.cos  / ) Large resolution ( R = m N ) Small free spectral range (FSR = /m) Small gap (d) gives: Problems as d  0 LIGO FPs: d  1 km FPIF

15. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ FP interference fringes Recall: Constructive (transmission) interference, or Destructive (reflection) interference when: 2  d.cos  = m Note cylindrical symmetry of fringes - cos  term - White = high intensity Black = low intensity

16. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ What you see with your eye Emission-line lab source (Ne, perhaps) – note the yellow fringes Orders: m (m-1) (m-2) (m-3) The central or “Jacquinot” spot

17. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ FP applications (traditional) Central (Jacquinot) spot scanning (0 th dimensional spectroscopy): Applicable for photo-multiplier where flux integrated over central spot Note: FP on optical axis The angular diameter ( J ) of FP central spot is given by the half-power points of the Airy Function when: sin(2  l.cos( J /2)/ ) =  /2 N This converts easily to: R = /  ~ 8/ J 2 or in other words, R  = 2  Let’s do some typical numbers: Say D T = 4m: d = 50mm, then for R ~20,000: J ~50 arcsec cf: Grating in Littrow condition: R = 2d.tan  /  D T With the same parameters, taking  = 30 º: Slit width  = 0.15 arcsec This is known (in some quarters) as the Jacquinot Advantage: … but why stop there?

18. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ FP with Area Detectors FP interferograms: Confocal with sky image Originally with photographic plates Good for line emission typically H  Fringe distortions due to changes in wavelength Used to map velocity fields in late-type galaxies Fundamental ambiguity: Is fringe distortion due to: a)Velocity field perturbation? b)Flux enhancement? c)or both?

19. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ Detection of feature in interferogram Is it a faint feature at 1 ? Is it a bright feature at 2 ? Radial intensity profile of FP interferogram Partial solution: Change the gap and repeat interferogram

20. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ Why only take 2 interferograms? Why not take 2*N? The birth of the TAURUS concept Note: A single interferogram only contains (1/ N ) th of in spatial information Scanning the FP across a full FSR is a natural way to: Recover ALL of the spatial information Resolve the fundamental ambiguity in the individual interferograms Scanning can be achieved in 3 ways: a)Changing  : - by tilting the FP or by moving the image across the fringes as in photographic interferograms (to partially resolve ambiguity) b)Changing  : - by changing the pressure of a gas in the FP gap as in 0 th dimensional scanning with eg: propane c)Scanning d: - by changing the physical gap between the plates very difficult since need to maintain parallelism to /2 N, at least TAURUS = QI etalons + IPCS … c1980

21. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ Queensgate Instruments (ET70)

22. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ CS100 controller FP etalons Queensgate Instruments (Capacitance Micrometry) Capacitors X-bridge Y-bridge Z-bridge (+ stonehenge) Piezo transducers (3) Construction Super-flat /200 base Centre piece (optical contact) Top plate Scanning d over 1*FSR m  (m-1) ; d  d + ( /2)

23. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ Alec Boksenberg and the Image Photon Counting System (IPCS) Imperial College, London  QI etalons + CS100 University College, London  IPCS TAURUS  16 Mbyte datacubes } Perfect synthesis …

24. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ How big a field? From before, Jacquinot central spot given by: R  = 2  or: R ~ 8  2 / 2 ~ 8  2 (d/D T ) 2 /  J 2, where  J is the angle on the sky. However, given an array detector, we can work off-axis: So, how far? Answer: Until the rings get narrower than  ~2 pixels (the seeing disk) Now, from the Airy Function we obtain: d  /d = -1/ 0.sin(  ) The full TAURUS field,  F is then:  F = 2  2 (d/D T ) 2 / R   F /  J ~ 20, typically or ~400 in 

25. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ TAURUS Datacube Recalling the phase delay equ n : m = 2  l.cos  For small values of  : goes as [1 – (  2 /2)] where   tan -1 [(y-y 0 ]/[x-x 0 ]) and (x 0,y 0 ) = centre of FP fringes  (x 1,y 1 ) is shifted in z-dir n w.r.t. (x 0,y 0 ) and this -shift is thus ~parabolic in  It is also periodic in “m”. We thus refer to this shift as a “Phase-correction” So the surface of constant is a “Nested Parabola” Cut through a “Nested Parabola and you get a set of rings and these rings are the FP fringes.

26. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ Wavelength Calibration (converting z to ) As shown, surfaces of constant, as seen in an (x,z)-slice are defined by a set of nested parabolæ, equally spaced in z. Any (x,y)-slice within the cube cuts through these nested paraboloids to give the familiar FP fringes (rings). Now -calibration requires transforming z  where: l (z) = l (0) + az a is a constant of proportionality. Constructive interference on axis (x 0,y 0 ) gives: az 0 = m 0 /2  - l (0) but an off-axis (x,y) point transmits the same 0 at (z 0 + p xy ) where: ap xy = l (z 0 ).(sec  xy – 1)

27. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ Phase Correction The 2D phase-map, p(x,y), can be defined such that: p(x,y) = m  z 0 (sec  – 1) The phase-map, p(x,y), can be obtained from a -calibration data-cube by illuminating the FP with a diffuse monochromatic source of wavelength,. at (x 2,y 2 ) z at (x 1,y 1 ) z I 0 (z) I 0 ( ) Phase corrected zz Note: Phase-map is discontinuous at each  z

28. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ The Phase-Map The phase-map is so called since it can be used to transform the raw TAURUS cube, with its strange multi-paraboloidal iso-wavelength contours into a well- tempered data cube where all (x,y)-slices are now at constant wavelengths. The process is called phase-correction since it represents a periodic function of period,  z. ie: If the z-value (z’) of a phase-shifted pixel exceeds the z-dimensions of the data-cube, then the spectra is simply folded back by one FSR to (z’ -  z). It will be noted that the phase-map (as defined previously) is independent of and hence in principle any calibration wavelength, cal, can be used to phase- correct a observation data-cube at an arbitary obs, remembering that:   2 &  z = /2 But also, the phase-map can be expressed in -space as:  xy = 0 (1 - cos  ) and hence is also independent of gap, l, and thus applicable to all FPs at all.

29. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ Order (m) and gap ( l ) determination The periodicity of the FP interference fringes makes -calibration non-trivial. The paraboloidal mapping from z  doesn’t exactly help, either! Nevertheless, using 2 calibration wavelengths: Say: 1 and 2, peak on-axis at z 1 and z 2 The trick is to find m1 (and hence m2), the order of interference. From m we can infer the gap, l, and hence obtain a -calibration where: az 0 = m 0 /2  - l (0) Then: zz z2 – z1 1 2 m1m1 2  ( 1  2 ) m2  m1m2  m1 = If m 1 can be estimated (from manufacturers specs or absolute capacitance measures) then we can search for a solution where m 2 is an integer. This can be an iterative process with several wavelength pairs. Clearly the further 1 and 2 are apart, the more accuracy is achieved.

30. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ Wavelength Calibration If m 1 can be estimated (from manufacturers specs or absolute capacitance measures) then we can search for a solution where m 2 is an integer. This can be an iterative process with several wavelength pairs. Clearly the further 1 and 2 are apart, the more accuracy is achieved. Once the interference order, m 1, for a known wavelength, 1, has been identified then wavelength calibration is given by: { } (z  z 0 ) m0z0m0z0 + 1 = 0

31. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ FP P in collimated beam Interference fringes formed at infinity: Sky and FP fringes are con-focal Detector sees FP fringes superimposed on sky image FPIF

32. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ FP I in image plane Interference fringes formed at infinity: Sky and FP fringes are not con-focal Detector sees FP plates superimposed on sky image ie: No FP fringes seen on detector FP is not perfectly centred on image plane (out of focus) to avoid detector seeing dust particles on plates. FPIF

33. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ FP (or Interference Filter) in image plane The FP still acts as a periodic monochromator but the angles into the FP (or IF) must not exceed the Jacquinot criteria, which states that: 2 < 8  2 / R (or R  = 2  ) At f/8: = 2.tan -1 (1/16) ~7.2º, so R < 500 At f/16: = 2.tan-1(1/32) ~3.6º, so R < 2,000 Note: the FoV is determined not by the width of the fringes but by the diameter of the FP. Also, an IF is simply a solid FP (  ~2.1) with very narrow gap.

34. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏

35. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ FB observations of NGC 7793 on the 3.60 m. Top left: DSS Blue Band image. Top right: Spitzer infrared array camera (IRAC) 3.6 μm image. Middle left: Hα monochromatic image. Middle right: Hα velocity field. Bottom: position-velocity (PV) diagram.

36. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ FB observations of NGC 7793 on the 36 cm. Top left: DSS Blue Band image. Top right: Spitzer IRAC 3.6 μm image. Middle left: Hα monochromatic image. Middle right: Hα velocity field. Bottom: PV diagram.

37. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ ADHOC screen shot (Henri)

38. Aug-Nov, 2008 IAG/USP (Keith Taylor) ‏ ADHOC screen shot (Henri)