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 – 1000

2. Supernova explosions:

Expansion velocities of ~ 3000 km/s

d/ = v/c = 3000/3x105 = 0.01

R > 100

R = 3.000

R = 30.000

35.0000.160100000

60.0000.09130000

100.0000.05310000

140.0000.046000

200.0000.0283000

Rth (Ang)T (K)

3. Thermal Broadening of Spectral lines:

3000001K

1000003G0

1200025F5

375080F0

2000150A0

R1Vsini (km/s)Sp. T.

4. Rotational Broadening:

1 2 pixel resolution, no other broadening

5. Chemical Abundances:

Hot Stars: R = 30.000

Cool Stars: R = 60.000 – 100.000

Driven by the need to resolve spectral lines and blends, and to accurately set the continuum.

6 Isotopic shifts:

Example:

Li7 : 6707.76

Li6 : 6707.92

R> 200.000

7 Line shapes (pulsations, spots, convection):

R=100.000 –200.000

Driven by the need to detect subtle distortions in the spectral line profiles.

collimator

Spectrographs

slit

camera

detector

corrector

From telescope

Anamorphic magnification:

d1 = collimator diameter

d2 = mirror diameter

r = d1/d2

slit

camera

detector

correctorFrom 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

without disperser

with disperser

with disperser

slit

fiber

Spectrographs are characterized by their angular dispersion

d

d

Dispersing element

ddA =

f

dl

dd

dld = f

In collimated light

S

dd

dld = S

In a convergent beam

Plate Factor

P = ( f A)–1

= ( f )–1

dd

P = ( f A)–1

= (S )–1

dd

P is in Angstroms/mm

P x CCD pixel size = Ang/pixel

w

h

f1

d1

A

D

f

d2

w´

h´

D = Diameter of telescope

d1 = Diameter of collimator

d2 = Diameter of camera

f = Focal length of telescope

f1 = Focal length of collimator

f2 = Focal length of collimator

A = Dispersing element

f2

w

h

f1

d1

A

D

d2

f

w´

h´

f2

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/f1

´= h/f1

w´ = rw(f2/f1) = rDF2

This expression is important for matching slit to detector:2 = rDF2 for Nyquist sampling (2 pixel projection of slit).1 CCD pixel () typically 15 – 20 m

Example 1:

= 1 arcsec, D = 2m, = 15m => rF2 = 3.1

Example 2:

= 1 arcsec, D = 4m, = 15m => rF2 = 1.5

Example 3:

= 0.5 arcsec, D = 10m, = 15m => rF2 = 1.2

Example 4:

= 0.1 arcsec, D = 100m, = 15m => rF2 = 0.6

5000 A

4000 An = –1

5000 A

4000 An = –2

4000 A

5000 An = 2

4000 A

5000 An = 1

Most of light is in n=0

b

The Grating Equation

m = sin + sin b 1/ = grooves/mm

dd =

m cos =

sin + sin cos

Angular Dispersion:

Linear Dispersion:

ddx

dd=

ddx

=1fcam

1

d/d

dx = fcam d

Angstroms/mm

Resolving Power:

w´ = rw(f2/f1) = rDF2

dx = f2 dd

f2 dd

rDF2

R = /d = Ar

1

d1

D

=rA

D

d1

For a given telescope depends only on collimator diameter

Recall: F2 = f2/d1

D(m) (arcsec) d1 (cm)

2 1 10

4 1 20

10 1 52

10 0.5 26

30 0.5 77

30 0.25 38

R = 100.000 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

d1

D=

Ar

d1

R d1

100000 0.6 cm

1000000 5.8 cm

For Peak efficiency the F-ratio (Focal Length / Diameter) of the telescope/collimator should be the same

collimator

1/F 1/F1

F1 = F

F1 > F

1/f is often called the numerical aperture NA

F1 < F

d/

1

But R ~ d1/

d1 smaller => must be smaller

Normal gratings:

• ruling 600-1200 grooves/mm

• Used at low blaze angle (~10-20 degrees)

• orders m=1-3

Echelle gratings:

• ruling 32-80 grooves/mm

• Used at high blaze angle (~65 degrees)

• orders m=50-120

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 = 2d1 tan D

A increases for increasing blaze angle

R2 echelle, tan = 2, = 63.4○

R4 echelle tan = 4, = 76○

At blaze peak + = 2

mb = 2 sin cos

b = blaze wavelength

3000

m=3

5000

m=2

4000 9000

m=1

6000 14000Schematic: 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=101 5000

5000 9000

9000 14000

Schematic: orders separated in the vertical direction for clarity

79 gr/mm grating

30002000

Need interference filters but why throw away light?

In reality:

collimator

Spectrographs

slit

camera

detector

corrector

From telescope

Cross disperser

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 ∞ –1

y

Cross dispersion

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 2Cross disperser

slit

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: 60.000 USD

Lens camera (choice): 1.000.000 USD

Reason: many elements (due to color terms), anti reflection coatings, etc.

Lost light

• Stability: Mechanical and Thermal?

HARPS

HARPS: 2.000.000 Euros

Conventional: 500.000 Euros

Tricks to improve efficiency:Overfill the Echelle

d1

d1

R ~ d1/

w´ ~ /d1

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 d1, d2, 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(2n)

Interferogram of a two frequency source:

I() = B1()cos(21) + B2(2)cos(22)

Interferogram of a two frequency source:

I() = Bi(i)cos(2i) = B()cos(2)d–∞

+∞

Inteferogram is just the Fourier transform of the brightness versus frequency, i.e spectrum