Spectrographs

Spectral Resolution d 1 2 Consider two monochromatic beams They will just be resolved when they have a wavelength separation of d Resolving power: d = full width of half maximum of calibration lamp emission lines R = d

Spectral Resolution The resolution depends on the science: 1. Active Galaxies, Quasars, high redshift (faint) objects: R = 500 – Supernova explosions: Expansion velocities of ~ 3000 km/s d / = v/c = 3000/3x10 5 = 0.01 R > 100

R = R =

R  th (Ang) T (K) 3. Thermal Broadening of Spectral lines:

K G F F A0 R1R1 Vsini (km/s)Sp. T. 4. Rotational Broadening: 1 2 pixel resolution, no other broadening

5. Chemical Abundances: Hot Stars: R = Cool Stars: R = – Driven by the need to resolve spectral lines and blends, and to accurately set the continuum.

6 Isotopic shifts: Example: Li 7 : Li 6 : R>

7 Line shapes (pulsations, spots, convection): R= – Driven by the need to detect subtle distortions in the spectral line profiles.

collimator Spectrographs slit camera detector corrector From telescope Anamorphic magnification: d 1 = collimator diameter d 2 = mirror diameter r = d 1 /d 2

slit camera detector corrector From telescope collimator Without the grating a spectograph is just an imaging camera

A spectrograph is just a camera which produces an image of the slit at the detector. The dispersing element produces images as a function of wavelength without disperser with disperser slit fiber

Spectrographs are characterized by their angular dispersion dd   d Dispersing element dd d A =

f dldl dd d dldl d = f In collimated light

S dd d dldl d = S In a convergent beam

Plate Factor P = ( f A ) –1 = ( f ) dd d P = ( f A ) –1 = ( S ) dd d P is in Angstroms/mm P x CCD pixel size = Ang/pixel

w h f1f1 d1d1 A D f d2d2   w´w´ h´h´  D = Diameter of telescope d 1 = Diameter of collimator d 2 = Diameter of camera f = Focal length of telescope f 1 = Focal length of collimator f 2 = Focal length of collimator A = Dispersing element f2f2

w h f1f1 d1d1 A D d2d2 f w´w´ h´h´ f2f2    w = slit width h = slit height Entrance slit subtends an angle  and  ´ on the sky:  = w/f  ´ = h/f Entrance slit subtends an angle  and  ´ on the collimator:  = w/f 1  ´ = h/f 1

w ´ = rw(f 2 /f 1 ) = r  DF 2 This expression is important for matching slit to detector: 2  = r  DF 2 for Nyquist sampling (2 pixel projection of slit). 1 CCD pixel (  ) typically 15 – 20  m Example 1:  = 1 arcsec, D = 2m,  = 15  m => rF 2 = 3.1 Example 2:  = 1 arcsec, D = 4m,  = 15  m => rF 2 = 1.5 Example 3:  = 0.5 arcsec, D = 10m,  = 15  m => rF 2 = 1.2 Example 4:  = 0.1 arcsec, D = 100m,  = 15  m => rF 2 = 0.6

5000 A 4000 A n = – A 4000 A n = – A 5000 A n = A 5000 A n = 1 Most of light is in n=0

bb   The Grating Equation m  = sin  + sin  b 1/  = grooves/mm 

dd d = m  cos  = sin  + sin  cos  Angular Dispersion: Linear Dispersion: d dx d dd = dd = 1 f cam 1 d  /d dx = f cam d  Angstroms/mm

Resolving Power: w ´ = rw(f 2 /f 1 ) = r  DF 2 dx = f 2 dd d  f 2 dd d  r  DF 2 R = / d = A r 1  d1d1 D  = rr A D d1d1 For a given telescope depends only on collimator diameter Recall: F 2 = f 2 /d 1

D(m)  (arcsec) d 1 (cm) R = A = 1.7 x 10 –3

What if adaptive optics can get us to the diffraction limit? Slit width is set by the diffraction limit:  = D R = r A D d1d1 D = A r d1d1 R d1d cm cm

For Peak efficiency the F-ratio (Focal Length / Diameter) of the telescope/collimator should be the same collimator 1/F 1/F 1 F 1 = F F 1 > F 1/f is often called the numerical aperture NA

F 1 < F d/1d/1 But R ~ d 1 /  d 1 smaller =>  must be smaller

Normal gratings: ruling grooves/mm Used at low blaze angle (~10-20 degrees) orders m=1-3 Echelle gratings: ruling grooves/mm Used at high blaze angle (~65 degrees) orders m= Both satisfy grating equation for = 5000 A

      Grating normal Relation between blaze angle , grating normal, and angles of incidence and diffraction Littrow configuration:  = 0,  =  =  m = 2  sin  A = 2 sin  R = 2d 1 tan  D A increases for increasing blaze angle R2 echelle, tan  = 2,  = 63.4 ○ R4 echelle tan  = 4,  = 76 ○ At blaze peak  +  = 2  m b = 2  sin  cos  b = blaze wavelength

3000 m= m= m= Schematic: orders separated in the vertical direction for clarity 1200 gr/mm grating 2 1 You want to observe 1 in order m=1, but light 2 at order m=2, where 1 ≠ 2 contaminates your spectra Order blocking filters must be used

4000 m=99 m=100 m= Schematic: orders separated in the vertical direction for clarity 79 gr/mm grating Need interference filters but why throw away light? In reality:

collimator Spectrographs slit camera detector corrector From telescope Cross disperser

yy ∞ 2 y m-2 m-1 m m+2 m+3   Free Spectral Range  m Grating cross-dispersed echelle spectrographs

Prism cross-dispersed echelle spectrographs yy ∞ –1 y

Cross dispersion yy ∞   · –1 = Increasing wavelength grating prism grism

Cross dispersing elements: Pros and Cons Prisms: Pros: Good order spacing in blue Well packed orders (good use of CCD area) Efficient Good for 2-4 m telescopes Cons: Poor order spacing in red Order crowding Need lots of prisms for large telescopes

Cross dispersing elements: Pros and Cons Grating: Pros: Good order spacing in red Only choice for high resolution spectrographs on large (8m) telescopes Cons: Lower efficiency than prisms (60-80%) Inefficient packing of orders

Cross dispersing elements: Pros and Cons Grisms: Pros: Good spacing of orders from red to blue Cons: Low efficiency (40%)

So you want to build a spectrograph: things to consider Chose R  product – R is determined by the science you want to do –  is determined by your site (i.e. seeing) If you want high resolution you will need a narrow slit, at a bad site this results in light losses Major consideration: Costs, the higher R, the more expensive

Chose  and , choice depends on – Efficiency – Space constraints – „Picket Fence“ for Littrow configuration

 normal 

White Pupil design? – Efficiency – Costs, you require an extra mirror

Tricks to improve efficiency: White Pupil Spectrograph echelle Mirror 1 Mirror 2 Cross disperser slit

Reflective or Refractive Camera? Is it fed with a fiber optic? Camera pupil is image of telescope mirror. For reflective camera: Image of Cassegrain hole of Telescope camera detector slit Camera hole Iumination pattern

Reflective or Refractive Camera? Is it fed with a fiber optic? Camera pupil is image of telescope mirror. For reflective camera: Image of Cassegrain hole camera detector A fiber scrambles the telescope pupil Camera hole ilIumination pattern

Cross-cut of illumination pattern For fiber fed spectrograph a refractive camera is the only intelligent option fiber e.g. HRS Spectrograph on HET: Mirror camera: USD Lens camera (choice): USD Reason: many elements (due to color terms), anti reflection coatings, etc. Lost light

Stability: Mechanical and Thermal? HARPS HARPS: Euros Conventional: Euros

Tricks to improve efficiency: Overfill the Echelle d1d1 d1d1 R ~ d 1 /  w´ ~  /d 1 For the same resolution you can increase the slit width and increase efficiency by 10-20%

Tricks to improve efficiency: Immersed gratings Increases resolution by factor of n n Allows the length of the illuminated grating to increase yet keeping d 1, d 2, small

Tricks to improve efficiency: Image slicing The slit or fiber is often smaller than the seeing disk: Image slicers reformat a circular image into a line

Fourier Transform Spectrometer

Interferogram of a monchromatic source: I(  ) = B( )cos(2  n  )

Interferogram of a two frequency source: I(  ) = B 1 (  )cos(2  1  ) + B 2 ( 2 )cos(2  2  )

Interferogram of a two frequency source: I(  ) =  B i ( i )cos(2  i  ) = B( )cos(2  )d  –∞ +∞ Inteferogram is just the Fourier transform of the brightness versus frequency, i.e spectrum