1. Spectroscopy principles
Jeremy Allington-Smith
University of Durham

2. Contents Reflection gratings in low order Grisms
Spectral resolution
Slit width issues
Grisms
Volume Phase Holographic gratings
Immersion
Echelles
Prisms
Predicting efficiency (semi-empirical)

3. Generic spectrograph layout
Camera
Collimator
Focal ratios defined as
Fi = fi / Di

4. Grating equation Interference condition: Grating equation: Dispersion:b A B A’ B’ n1 n2
Interference condition:
 path difference between AB and A'B'
Grating equation:
Dispersion: f2 dx db

5. "Spectral resolution" Terminology (sometimes vague!)
Wavelength resolution dl
Resolving power
Classically, in the diffraction limit,
Resolving power = total number of rulings x spectral order
I.e.
But in most practical cases for astronomy (c < l/DT), the resolving power is determined by the width of the slit, so R < R* dl l
Total grating length

6. Spectral resolution Spectral resolution: Projected slit width:
Conservation of Etendue (nAW)
  
Image of slit
on detector
Camera focal
length

7. Resolving power Illuminated grating length:
Spectral resolution (width)
Resolving power:
expressed in laboratory terms
expressed in astronomical terms
since and
Collimator
focal ratio
Physical
slitwidth
Size of spectrograph must scale with telescope size
Grating
length
Angular
slitwidth
Telescope
size

8. Importance of slit width
Width of slit determines:
Resolving power (R) since Rc = constant
Throughput (h)
Hence there is always a tradeoff
between throughput and spectral information
Function h(c) depends on Point Spread Function (PSF) and profile of extended source
generally h(c) increases slower than c+1 whereas R  c-1 so hR maximised at small c
Signal/noise also depends on slit width
throughput ( signal)
wider slit admits more sky background ( noise)

9. 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

10. Anamorphism Beam size in dispersion direction:
Output
angle
Beam size in dispersion direction:
Beam size in spatial direction:
Anamorphic factor:
Ratio of magnifications:
if b < a, A > 1, beam expands
W increases  R increases
image of slit thinner  oversampling worse
if b > a, A < 1, beam squashed
W reduces  R reduces
image of slit wider  oversampling better
if b = a, A = 1, beam round
Littrow configuration
Input
angle
dispersion

11. Generic spectrograph layout
Camera
Collimator
Fi = fi / Di

12. Blazing Diffracted intensity: Shift envelope peak to m=1
b = active width
of ruling (b  a)
Diffracted intensity:
Shift envelope peak to m=1
Blaze condition
specular reflection off grooves:
also 
since
Interference
pattern
Single slit
diffraction
F = phase difference between adjacent rulings
q = phase difference from centre of one ruling to its edge

13. Efficiency vs wavelength
Approximation valid for a > l
lmax(m) = lB(m=1)/m
Rule-of-thumb:
40.5% x peak at
 (large m)
Sum over all orders < 1
reduction in efficiency with increasing order 2 3 4 6 5
(See: Schroeder, Astronomical Optics)

14. Don't forget higher orders!
Order overlaps
Effective passband
in 1st order
Don't forget higher orders!
Intensity
1st order
blaze profile
m=1
First and second
orders overlap!
m=2
Passband
in 2nd order
Zero order
matters for MOS
2nd order
blaze profile
Passband
in zero
order
m=0
Wavelength in first order marking position on detector in dispersion direction (if dispersion ~linear)
1st order lL lC
2lL lU
2lU
(2nd order) lL lU

15. Order overlaps To eliminate overlap between 1st and 2nd order
1st order
Detector
Zero
order
2nd order
dispersion
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 2lL = lU)
Zero order may be a problem in multiobject spectroscopy

16. 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 (Y = 0)
convert by multiplying wavelength by cos(Y/2)
Coating may affect grating properties in complex way for large g (don't scale just by reflectivity!)
Two prediction software tools on market
differential
integral

17. GMOS optical system Detector (CCD mosaic) Masks and
Integral Field Unit
Detector (CCD
mosaic)
From
telescope
CCD mosaic
(6144x4608)
Mask field (5.5'x5.5')
Science fold
mirror field (7')

18. Example of performance
GMOS grating set
D1 = 100mm, Y = 50
DT = 8m, c = 0.5"
m = 1, 13.5mm/px
Intended to overcoat all with silver
Didn't work for those with large groove angle - why?
Actual blaze curves differed from scalar theory predictions

19. Grisms Transmission grating attached to prism
Allows in-line optical train:
simpler to engineer
quasi-Littrow configuration - no variable anamorphism
Inefficient for r > 600/mm due to groove shadowing and other effects

20. Grism equations Modified grating equation: Undeviated condition:
n'= 1, b = -a = f
Blaze condition: q = 0  lB = lU
Resolving power
(same procedure as for grating)
q = phase difference from centre of one ruling to its edge

21. 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:
Don't confuse with 'holographic' gratings (SR)
Advantages:
Higher peak efficiency than SR
Possibility of very large size with high r
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

22. 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

23. VPH equations Modified grating equation: Blaze condition:
= Bragg diffraction
Resolving power:
Tune blaze condition by tilting grating (a)
Collimator-camera angle must also change by 2a  mechanical complexity

24. VPH efficiency Kogelnik's analysis when: Bragg condition when:
Bragg envelopes (efficiency FWHM):
in wavelength:
in angle:
Broad blaze requires
thin DCG
large index amplitude
Superblaze
Barden et al. PASP 112, 809 (2000)

25. VPH 'grism' = vrism Remove bent geometry, allow in-line optical layout
Use prisms to bend input and output beams while generating required Bragg condition

26. Limits to resolving power
Resolving power can increase as m, r and W increase for a given wavelength, slit and telescope
Limit depends on geometrical factors only - increasing r 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 diffraction-limited regime is entered (l > cDT)
Grating
parameters
Geometrical
factors

27. Limits with normal gratings
For GMOS with c = 0.5", DT = 8m, D1 =100mm, Y =50
R and l plotted as function of a
A(max) = 1.5 since
D2(max) = 150mm  R(max) ~ 5000
Normal SR
gratings
Simultaneous
l range

28. 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!
Potential drawbacks:
loss of efficiency
ghost images
but Lee & Allington-Smith (MNRAS, 312, 57, 2000) show this is not the case

29. Limits with immersed gratings
For GMOS with c = 0.5", DT = 8m, D1 = 100mm
R and l plotted as function of a
With immersion R ~ okay with wide slit
Immersed
gratings

30. Echelle gratings Obtain very high R (> 105) using very long grating
In Littrow:
Maximising g requires large mr since mrl = 2sing
Instead of increasing r, increase m
Echelle is a coarse
grating with large
groove angle
R parameter = tang
(e.g R2  g = 63.5)
Groove
angle

31. Multiple orders Many orders to cover desired ll: Free spectral range
Dl = l/m
Orders lie on top of each other:
l(m) = l(n) (n/m)
Solution:
use narrow passband filter to isolate one order at a time
cross-disperse to fill detector with many orders at once
Cross dispersion may use prisms or low dispersion grating

32. Echellette example - ESI
Sheinis et al. PASP 114, 851 (2002)

33. 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

34. Dispersion for prisms
Fermat's principle:
Dispersion:

35. Resolving power for prisms
Angular width
of resolution
element
on detector
Basic definitions:
Conservation of Etendue:
Result:
Comparison of grating and prism:
Angular
dispersion
Angular
slitwidth
Beam
size
Telescope
aperture
Disperser
'length'
'Ruling
density'

36. * ESO/LAM/Durham/Astrium et al. for ESA
Prism example
A design for Near-infrared spectrograph* of NGST
DT = 8m, c = 0.1", D1 = D2 = 86mm, 1 < l < 5mm
R  100 required
Raw refractive index data for sapphire
Collimator
Slit plane
Double-pass prism+mirror
Detector
Camera
* ESO/LAM/Durham/Astrium et al. for ESA

37. Prism example (contd) Required prism thickness,t:
sapphire: 20mm
ZnS/ZnSe: 15mm
Uniformity in dl or R required?
For ZnS:
n  2.26  a = 75.3
f = 12.9

38. Appendix: Semi-empirical efficiency prediction for classical gratings

39. Efficiency - semi-empirical
Efficiency as a function of rl depends mostly on g
Different behaviour depends on polarisation:
P - parallel to grooves (TE)
S - perpendicular to grooves (TM)
Overall peak at rl = 2sing (for Littrow examples)
Anomalies (passoff) when light diffracted from an order at b = p/2  light redistributed into other orders
discontinuities at (Littrow only)
Littrow: symmetry m 1-m
Otherwise: no symmetry (rl depends on m,Y)  double anomalies
Also resonance anomalies - harder to predict

40. Efficiency - semi-empirical (contd)
Different regimes for blazed (triangular) grooves
g < 5 obeys scalar theory, little polarisation effect (P  S)
5 < g < 10 S anomaly at rl  2/3 , P peaks at lowerrl than S
10 < g < 18 various S anomalies
18 < g < 22 anomalies suppressed, S >> P at large rl
22 < g < 38 strong S anomaly at P peak, S constant at large rl
g > 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, ( rl
a=b