Transcript Spectrographs-lecture-2-Tautenburg.ppt

Spectrographs
Spectral Resolution
dl
Consider two monochromatic
beams
They will just be resolved when
they have a wavelength
separation of dl
Resolving power:
l1
l2
l
R=
dl
dl = full width of half
maximum of calibration
lamp emission lines
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
dl/l = v/c = 3000/3x105 = 0.01
R > 100
R = 30.000
R = 3.000
3. Thermal Broadening of Spectral lines:
T (K)
Dlth (Ang)
R
3000
0.028
200.000
6000
0.04
140.000
10000
0.053
100.000
30000
0.091
60.000
100000
0.160
35.000
4. Rotational Broadening:
Sp. T.
1
Vsini (km/s)
R1
A0
150
2000
F0
80
3750
F5
25
12000
G0
3
100000
K
1
300000
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.
Spectrographs
Anamorphic magnification:
d1 = collimator diameter
d2 = mirror diameter
r = d1/d2
camera
detector
corrector
From telescope
slit
collimator
From telescope
collimator
corrector
slit
Without the grating a spectograph
is just an imaging camera
camera
detector
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
slit
without
disperser
with disperser
fiber
with disperser
without
disperser
Spectrographs are characterized by their angular dispersion
Dispersing element
b
l
l + dl
db
A
=
db
dl
In collimated
light
f
dl
dl
dl
=
f
db
dl
In a convergent beam
S
dl
dl
=
S
db
dl
Plate Factor
–1
P = (f A) = (f
db
dl
–1
P = (f A) = (S
db
dl
–1
)
–1
)
P is in Angstroms/mm
P x CCD pixel size = Ang/pixel
f
da
d1
w
d2
A
db
D
h´
h
f
f1
f2
D = Diameter of telescope
f = Focal length of telescope
d1 = Diameter of collimator
f1 = Focal length of collimator
d2 = Diameter of camera
f2 = Focal length of collimator
A = Dispersing element
w´
f
da
d1
w
A
D
w = slit width
h = slit height
db
w´
h´
h
f
d2
f1
f2
Entrance slit subtends an
angle f and f´on the sky:
Entrance slit subtends an angle
da and da´on the collimator:
f = w/f
da = w/f1
f´= h/f
da´= h/f1
w´ = rw(f2/f1) = rfDF2
This expression is important for matching slit to detector:
2D = rfDF2 for Nyquist sampling (2 pixel projection of slit).
1 CCD pixel (D) typically 15 – 20 mm
Example 1:
f = 1 arcsec, D = 2m, D= 15mm => rF2 = 3.1
Example 2:
f = 1 arcsec, D = 4m, D= 15mm => rF2 = 1.5
Example 3:
f = 0.5 arcsec, D = 10m, D= 15mm => rF2 = 1.2
Example 4:
f = 0.1 arcsec, D = 100m, D= 15mm => rF2 = 0.6
5000 A
n = –2
4000 A
5000 A
4000 A
4000 A
5000 A
n = –1
Most of
light is in
n=0
n=1
4000 A
n=2
5000 A
The Grating Equation
s
bb
f
a
ml
s =
sin a + sin bb
1/s = grooves/mm
Angular Dispersion:
m
db
= s cos b =
dl
Linear Dispersion:
sin a + sin b
l cos b
dx = fcam db
dl
dl db
=
=
dx
db dx
1
1
fcam db/dl
Angstroms/mm
Resolving Power:
dx = f2 db Dl
dl
w´ = rw(f2/f1) = rfDF2
f2
Recall: F2 = f2/d1
db
Dl = rfDF2
dl
rf
dl =
A
R = l/dl =
D
d1
For a given telescope
depends only on collimator
diameter
lA 1
r
f
d1
D
A = 1.7 x 10–3
R = 100.000
D(m)
2
4
10
10
30
30
f (arcsec)
1
1
1
0.5
0.5
0.25
d1 (cm)
10
20
52
26
77
38
What if adaptive optics can get us to the diffraction
limit?
Slit width is set by the diffraction limit:
f= l
D
l A D
R=
r
l
R
100000
1000000
d1
A
d1
=
D
r
d1
0.6 cm
5.8 cm
For Peak efficiency the F-ratio (Focal Length / Diameter) of the
telescope/collimator should be the same
1/F
1/F1
F1 = F
F1 > F
1/f is often called
the numerical
aperture NA
d /1
F1 < F
But R ~ d1/f
d1 smaller => f 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 l = 5000 A
a
b
q
d
d
q
Grating normal
Relation between blaze angle d,
grating normal, and angles of
incidence and diffraction
Littrow configuration:
m l = 2 s sin d
q = 0, a = b = d
A = 2 sin d/l
A increases
for increasing
blaze angle
R = 2d1 tan d/f D
R2 echelle, tan d = 2, d = 63.4○
R4 echelle tan d = 4, d = 76○
At blaze peak a + b = 2d
mlb = 2 s sin d cos q
lb = blaze wavelength
1200 gr/mm grating
Schematic: orders separated in the
vertical direction for clarity
m=1
l1
6000
14000
m=2
4000
m=3
9000
l2
3000
5000
You want to observe l1 in order m=1, but light l2 at order m=2,
where l1 ≠ l2 contaminates your spectra
Order blocking filters must be used
79 gr/mm grating
In reality:
Schematic: orders separated in the
vertical direction for clarity
9000
14000
m=99
m=100
m=101
5000
9000
4000
5000
2000
3000
Need interference filters but why throw away light?
Spectrographs
camera
detector
corrector
Cross disperser
From telescope
slit
collimator
dl
Free Spectral Range Dl = l/m
m-2
m-1
m
m+2
m+3
y
Dy ∞ l2
Grating cross-dispersed echelle spectrographs
Prism cross-dispersed echelle spectrographs
y
Dy ∞ l–1
Cross dispersion
grism
prism
grating
Increasing
wavelength
Dy ∞ l2 · l–1 = l
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 Rf product
– R is determined by the science you want to do
– f 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 q and g, choice depends on
– Efficiency
– Space constraints
– „Picket Fence“ for Littrow configuration
normal
a
g
• 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
slit
• Reflective or Refractive Camera? Is it fed with a fiber
optic?
Camera pupil is image of telescope mirror. For reflective camera:
slit
camera
Image of
Cassegrain
hole of
Telescope
detector
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:
A fiber scrambles the telescope pupil
camera
Image of
Cassegrain
hole
detector
Camera
hole
ilIumination
pattern
Cross-cut of illumination pattern
fiber
Lost light
For fiber fed spectrograph a refractive camera is the only
intelligent option
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.
• Stability: Mechanical and Thermal?
HARPS
HARPS: 2.000.000 Euros
Conventional: 500.000 Euros
Tricks to improve efficiency:
Overfill the Echelle
d1
R ~ d1/f
w´ ~ f/d1
d1
For the same resolution you
can increase the slit width
and increase efficiency by
10-20%
Tricks to improve efficiency:
Immersed gratings
n
Increases resolution
by factor of 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(d) = B(n)cos(2pnd)
Interferogram of a two frequency source:
I(d) = B1(n1)cos(2pn1d) + B2(n2)cos(2pn2d)
Interferogram of a two frequency source:
+∞
I(d) = S Bi(ni)cos(2pnid) =
 B(n)cos(2pnd)dn
–∞
Inteferogram is just the Fourier transform of the brightness
versus frequency, i.e spectrum