Spectrographs

Literature: Astronomical Optics, Daniel Schneider Astronomical Observations, Gordon Walker Stellar Photospheres, David Gray

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

R = R = R =

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.

Line shapes due to Convection Hot rising cell Cool sinking lane The integrated line profile is distorted. Amplitude of distortions ≈ 10s m/s

R = R >

8 Stellar Radial Velocities:  RV (m/s) ~ R –3/2 (  ) –1/2  wavelength coverage R  (m/s)

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 h ´ = h(f 2 /f 1 ) =  ´ DF 2 F 2 = f 2 /d 1 r = anamorphic magnification due to dispersing element = d 1 /d 2

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