Transcript No Slide Title

Spectroscopy principles
Jeremy Allington-Smith
University of Durham
Contents
• Reflection gratings in low order
– Spectral resolution
– Slit width issues
•
•
•
•
•
•
Grisms
Volume Phase Holographic gratings
Immersion
Echelles
Prisms
Predicting efficiency (semi-empirical)
Generic spectrograph layout
f2
Focal ratios
defined as
Fi = fi / Di
Detector
Camera
fT
D2
Grating
Slit
DT

s



Telescope

D1
W
f1
Collimator
Grating equation
n1
n2
A
• Interference condition:
B

 path difference between AB and A'B'

A’
B’
• Grating equation:
m  n1 sin   n2 sin 
a
where
1

a
• Dispersion:
d cos 

d
m
d d d cos 


dx d dx m f 2
f2
d
dx
"Spectral resolution"
• Terminology (sometimes vague!)
– Wavelength resolution d
– Resolving power R   d
• Classically, in the diffraction limit,
Resolving power = total number of rulings
x spectral order
*
I.e. R  mW
Total grating length
• But in most practical cases for astronomy
( < /DT), the resolving power is determined
by the width of the slit, so R < R*
d

Spectral resolution
• Spectral resolution:
cos 
 d 
d   s' 
s'
m f 2
 dx 
• Projected slit width:
f1
Conservation of Etendue (nAW)
s
F2




s


s

 s'  s  s
'
F1
F
sD cos 
 d   d s'  cos  s 2  1
 dx 
m f 2
F1
Image of slit
on detector
mD2 f1
Camera focal
length

slit
f2
D1
D2
'
s'
Image
of slit
Resolving power
• Illuminated grating length:
D2
W
cos 
• Spectral resolution (width)
s
d 
mF1W
• Resolving power:
Collimator
focal ratio
– expressed in laboratory terms
R
 mF1W

d
s
– expressed in astronomical terms
R
mW
DT
since
s   fT
and
fT
f
 1  FT  F1
DT D1
Physical
slitwidth
Grating
length
Angular Telescope
slitwidth
size
Size of spectrograph must scale with telescope size
Importance of slit width
• Width of slit determines:
– Resolving power (R) since R = constant
– Throughput (h)
• Hence there is always a tradeoff
between throughput and spectral information
• Function h() depends on Point Spread Function (PSF)
and profile of extended source
– generally h() increases slower than 1 whereas R  -1
so hR maximised at small 
• Signal/noise also depends on slit width
– throughput ( signal)
– wider slit admits more sky background ( noise)
Signal/noise vs slit width
• For GTC/EMIR in K-band (Balcells et al. 2001)
SNR falls as slit
includes more
sky background
Optimum
slit width
Anamorphism
Output
angle
• Beam size in dispersion direction:
D2  W cos 
• Beam size in spatial direction:
D1  W cos
D
cos 
A 2 
D1 cos
M x D2

A
M  D1
• Anamorphic factor:
• Ratio of magnifications:
– if  < , A > 1, beam expands
• W increases  R increases
• image of slit thinner  oversampling
worse
– if  > , A < 1, beam squashed
Input
angle
D1
D2
D1
D2
• W reduces  R reduces
• image of slit wider  oversampling better
– if  = , A = 1, beam round
• Littrow configuration
Input
Output
dispersion
Generic spectrograph layout
f2
Fi = fi / Di
Detector
Camera
fT
D2
Grating
Slit
DT

s



Telescope

D1
W
f1
Collimator
Blazing
b = active width
of ruling (b  a)
• Diffracted intensity:
Single slit
diffraction
 sin 2 N  sin 2 

I  
2
2
 sin   



/b
Intensity
Interference
pattern
Blaze function
/a
F  phase difference between adjacent rulings
 = phase difference from centre of one ruling to its edge
-1
0
Spectral order, m
Facet
normal
 Shift envelope peak to m=1


• Blaze condition
i
specular reflection off grooves:
    2 also    - 
sin x  sin y  2 sin
x y
x- y
cos
2
2

r

 m  sin   sin   2 sin  cos 
B
since
1
2
Grating
normal
2
Efficiency
Efficiency vs wavelength
• Approximation valid for a > 
 max(m) = B(m=1)/m
• Rule-of-thumb:
40.5% x peak at
 - - 
m
B
1
(large m)
– reduction in efficiency with
increasing order
0 .8
2
3
4
6 5
0 .9
• Sum over all orders < 1
Wavelength
m e d iu m re s o lu tio n g ra tin g o rd e rs 2 -6
0 .7
Efficie ncy

m=2
IS A A C g ra tin g e ffie n c y (fro m E S O E T C )
2m B
2m B
 
and  - 
2m - 1
2m  1
B
m=1
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0
0 .5
1
1 .5
2
W a v e le n g t h ( u m )
(See: Schroeder, Astronomical Optics)
2 .5
3
Order overlaps
Effective passband
in 1st order
Don't forget
higher orders!
Intensity
1st order
blaze profile
m=1
m=2
Passband
in 2nd order
0
(2nd order)
0
L
Zero order
matters
for MOS
2nd order
blaze profile
Passband
in zero
order
m=0
1st order
First and second
orders overlap!
C 2L U
L
2U
U
Wavelength in first
order marking
position on detector
in dispersion direction
(if dispersion ~linear)
Order overlaps
2nd order
Detector
dispersion
Zero
order
1st order
To eliminate overlap between 1st and 2nd order
• Limit wavelength range incident on detector using
passband filter or longpass ("order rejection") filter
acting with long-wavelength cutoff of optics or
detector (e.g. 1100nm for CCD)
• Optimum wavelength range is 1 octave (then 2L = U)
• Zero order may be a problem in multiobject spectroscopy
Predicting efficiency
• Scalar theory approximate
– optical coating has large and unpredictable effects
– grating anomalies not predicted
– Strong polarisation effect at high ruling density
(problem if source polarised or for spectropolarimetry)
• Fabricator's data may only apply to Littrow (  0)
– convert by multiplying wavelength by cos(/2)
– grating anomalies not predicted
• Coating may affect grating properties in complex
way for large  (don't scale just by reflectivity!)
• Two prediction software tools on market
– differential
– integral
CCD mosaic
(6144x4608)
GMOS optical system
Mask field (5.5'x5.5')
Detector (CCD
mosaic)
Science fold
mirror field (7')
Masks and
Integral Field Unit
From
telescope
Example of performance
•
Absolute efficiency of fitted GMOS gratings
(Littrow, unpolarised)
GMOS grating set
– D1 = 100mm,  = 50
•
•
– DT = 8m,  = 0.5"
90
– m = 1, 13.5mm/px
70
Intended to overcoat
all with silver
80
Efficiency (%)
•
100
R831
60
B600
50
R600
40
R400
30
R150
20
Didn't work for those
with large groove
angle - why?
Actual blaze curves
differed from scalar
theory predictions
B1200
10
0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Wavelength (microns)
GMOS
grating
name
B1200
R831
B600
R600
R400
R150
Coating
Al
Al
Ag
Al
Ag
Ag
Ruling
density,
 (mm-1)
1200
831
600
600
400
150
Groove
Nominal
angle, 
RB
(deg) B(nm)
17.5
463
3744
19.7
757
4396
8.6
461
1688
17.5
926
3744
9.7
764
1918
3.4
717
631
Actual
B(nm)
490
750-900
540
950
940
670-740
Dispersion,
d/dx
(nm/pixel)
0.023
0.034
0.047
0.047
0.085
0.174
Grisms
• Transmission grating
attached to prism
• Allows in-line optical
train:
– simpler to engineer
– quasi-Littrow
configuration - no
variable anamorphism
• Inefficient for
 > 600/mm due to
groove shadowing
and other effects



D1
d

nG
nR
n'
Grism equations
• Modified grating equation:
m  n sin   n' sin 
• Undeviated condition:
mU  (n - 1) sin 
n'= 1,   -  
 = phase difference from
centre of one ruling to its edge
• Blaze condition:
  0  B = U
• Resolving power
R
mW
DT
W
D1
(same procedure as for grating)
cos
(n - 1) tan D1
R
DT
Volume Phase Holographic gratings
• So far we have considered surface relief gratings
• An alternative is VPH in which refractive index
varies harmonically throughout the body of the
grating: ng ( x, z)  ng  ng cos2 g (x sin   z cos )
• Don't confuse with 'holographic' gratings (SR)
• Advantages:
–
–
–
–
Higher peak efficiency than SR
Possibility of very large size with high 
Blaze condition can be altered (tuned)
Encapsulation in flat glass makes more robust
• Disadvantages
– Tuning of blaze requires bendable spectrograph!
– Issues of wavefront errors and cryogenic use
VPH configurations
• Fringes = planes of
constant n
• Body of grating made
from Dichromated
Gelatine (DCG) which
permanently adopts
fringe pattern generated
holographically
• Fringe orientation allows
operation in transmission
or reflection
  /2



2-2


VPH equations
• Modified grating equation:
m  sin   sin 
• Blaze condition:
mB  2ng sin  g  2 sin 
= Bragg diffraction
• Resolving power:
• Tune blaze condition by
tilting grating ()
• Collimator-camera angle
must also change by 2
 mechanical complexity
ng sin  g  sin 
mW m D1
R

DT
DT cos
VPH efficiency
• Kogelnik's analysis when:
2d g2
• Bragg condition when:
• Bragg envelopes (efficiency FWHM):
n g d . 
ng
. > 10

2




1
1




n g 
– in wavelength:   



  g tan g 
  g tan g  d
– in angle:
 
1
gd
• Broad blaze requires
– thin DCG
– large index amplitude
• Superblaze
Barden et al. PASP 112, 809 (2000)
VPH 'grism' = vrism
• Remove bent geometry, allow in-line optical layout
• Use prisms to bend input and output beams while
generating required Bragg condition
n1

ng
n1
d
D1

R
mW m

D1 (1  tand tan )
DT
DT
 sin  

 n1 
d   - arcsin
Limits to resolving power
• Resolving power can increase as m,  and W
increase for a given wavelength, slit and telescope
Grating
parameters
R
D
mW
 1
DT
DT
 sin   sin  


cos 


Geometrical
factors
• Limit depends on geometrical factors only increasing  or m will not help!
• In practice, the limit is when the output beam
overfills the camera:
– W is actually the length of the intersection between
beam and grating plane - not the actual grating length
– R will increase even if grating overfilled until diffractionlimited regime is entered ( > DT)
Limits with normal gratings
• For GMOS with
 = 0.5", DT = 8m,
D1 =100mm, 
=50
• R and  plotted as
function of 
•
A(max) = 1.5 since
D2(max) = 150mm
 R(max) ~ 5000
Grating
A
B
C
D
H
IA
X1
X2
X3
 (mm )
 (deg)
1200
831
600
600
400
158
2400
1800
1200
17.5
19.7
8.6
17.5
9.7
3.6
21.0
26.7
17.5
-1
 (deg)
0.0
0.0
0.0
0.0
0.0
0.0
30.0
30.0
30.0
B
463
757
461
926
764
720
444
724
724
RB
3744
4396
1688
3744
1918
668
7043
9572
5372
Normal SR
gratings
Simultaneous
 range
Immersed gratings
• Beat the limit using a prism to squash the output
beam before it enters the camera:
 D2 kept small while W can be large
• Prism is immersed to prism using an optical
couplant (similar n to prism and high transmission)
• For GMOS R(max) ~ doubled!
camera
• Potential drawbacks:
– loss of efficiency
– ghost images
– but Lee & Allington-Smith
(MNRAS, 312, 57, 2000)
show this is not the case
prism
collimator
grating
Immersed 1800/mm grating
with 35 deg prism, 630-682nm
Limits with immersed gratings
• For GMOS with
 = 0.5", DT = 8m,
D1 = 100mm
• R and  plotted as
function of 
• With immersion
R ~ 10000 okay
with wide slit
Grating
A
B
C
D
H
IA
X1
X2
X3
 (mm-1)
 (deg)
1200
831
600
600
400
158
2400
1800
1200
17.5
19.7
8.6
17.5
9.7
3.6
21.0
26.7
17.5
 (deg)
0.0
0.0
0.0
0.0
0.0
0.0
30.0
30.0
30.0
B
463
757
461
926
764
720
444
724
724
RB
3744
4396
1688
3744
1918
668
7043
9572
5372
Immersed
gratings
Echelle gratings
• Obtain very high R (> 105) using very long grating
• In Littrow:
R
mW 2 D1

tan
DT
DT
Groove
angle
• Maximising  requires large m since m = 2sin
• Instead of increasing , increase m
• Echelle is a coarse
grating with large
groove angle
• R parameter = tan
(e.g R2   = 63.5)

D1

W
Multiple orders
– use narrow passband
filter to isolate one
order at a time
– cross-disperse to fill
detector with many
orders at once
10
555
500
11
Cross-dispersion
• Many orders to
cover desired :
Free spectral range
  /m
• Orders lie on top of
each other:
(m) = (n) (n/m)
• Solution:
555
454
12
417
13
385
14
357
15
333
16
333
312
312
300
Primary dispersion
Cross dispersion may use prisms
or low dispersion grating
Echellette example - ESI
Sheinis et al. PASP 114, 851 (2002)
Prisms
• Useful where only low resolving power is required
• Advantages:
– simple - no rulings! (but glass must be of high quality)
– multiple-order overlap not a problem - only one order!
• Disadvantages:
– high resolving power not possible
– resolving power/resolution can vary strongly with 

 
L
D1
D2
t
Prism, index n
Dispersion for prisms
• Fermat's principle:
• Dispersion:
tn  2L cos 
d d dn

d dn d
d
d
 -2
dn
dn
   -  - 2
dn - 2 L sin  - 2 D1


d
t
t
dn
1 dn D1

d
2 d
t

 
L
D1
D2
t
Prism, index n
d D1 d

d
t dn
Resolving power for prisms
 d 
d   d
 d 
• Basic definitions:
• Conservation of Etendue:
Angular
slitwidth
DT  d D2
Telescope
aperture
• Result:
Angular width
of resolution
element
on detector
Beam
size
R
• Comparison of grating and prism:
Disperser
'length'

d
Angular
dispersion
 D d  DT  DT d
 
d   1

t dn
 t dn  D2 
R *prism  t
R
dn
d
*
Rgrat
 mW
'Ruling
density'
 dn  
t
 
DT d  DT
 *
 R prism

Prism example
A design for Near-infrared spectrograph* of NGST
• DT = 8m,  = 0.1", D1 = D2 = 86mm, 1 <  < 5mm
• R  100 required
Raw refractive index
Slit plane
data for sapphire
Collimator
2.3
0
-0.02
2.29
n
-0.06
n (crystan)
2.28
-0.08
dn/dlambda
(Crystan)
2.27
-0.1
-0.12
Detector
2.26
Camera
dn/dlambda
Double-pass
prism+mirror
-0.04
-0.14
-0.16
2.25
-0.18
2.24
-0.2
1
2
3
4
5
Wavelength (um)
* ESO/LAM/Durham/Astrium et al. for ESA
Prism example
(contd)
• Required prism
thickness,t:
– sapphire: 20mm
– ZnS/ZnSe: 15mm
• Uniformity in d
or R required?
• For ZnS:
 2 D1 

 nt 
n  2.26   = 75.3
t
t
sin  

sin 
2

 2 L 2 D1
 = 12.9
1000
  arctan
900
R
L  D1 sin 
Resolving power
800
R for t=20mm ZnSe
700
R for t=20mm ZnS
600
R for t=15mm Sappphire
500
Fit for ZnS
400
Fit for sapphire
300
200
100
0
1
2
3
Wavelength (m m)
4
5
Appendix: Semi-empirical
efficiency prediction for
classical gratings
Efficiency - semi-empirical
• Efficiency as a function of  depends mostly on 
• Different behaviour depends on polarisation:
P - parallel to grooves (TE)
S - perpendicular to grooves (TM)
• Overall peak at  = 2sin (for Littrow examples)
• Anomalies (passoff) when light diffracted from an
order at  = /2  light redistributed into other orders
2
– discontinuities at a 
2m - 1
• Littrow: symmetry m 1-m
(Littrow only)
m
 a
-3
0.29
-2
0.40
-1
0.67
0
-
+1
-
+2
0.67
• Otherwise: no symmetry ( depends on m,)  double anomalies
– Also resonance anomalies - harder to predict
+3
0.40
Efficiency - semi-empirical
(contd)
Different regimes for blazed (triangular) grooves
 < 5
obeys scalar theory, little polarisation effect (P  S)
5 <  < 10
S anomaly at   2/3 , P peaks at lower than S
10 <  < 18
various S anomalies
18 <  < 22
anomalies suppressed, S >> P at large 
22 <  < 38
strong S anomaly at P peak, S constant at large 
 > 38
S and P peaks very different, efficient in Littrow only
NOTE
Results apply to
Littrow only
From: Diffraction
Grating Handbook,
C. Palmer, Thermo RGL,
(www.gratinglab.com)
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