Spectroscopy principles

1
Spectroscopy principles

• Jeremy Allington-Smith
• University of Durham

2
Contents

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

3
Generic spectrograph layout
Focal ratios defined as Fi fi / Di
4
Grating equation

• 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 lt
  l/DT), the resolving power is determined by the
  width of the slit, so R lt R

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
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 c1 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
Output angle

• Beam size in dispersion direction
• Beam size in spatial direction
• Anamorphic factor
• Ratio of magnifications
• if b lt a, A gt 1, beam expands
• W increases ? R increases
• image of slit thinner ? oversampling worse
• if b gt a, A lt 1, beam squashed
• W reduces ? R reduces
• image of slit wider ? oversampling better
• if b a, A 1, beam round
• Littrow configuration

Input angle
11
Generic spectrograph layout
Fi fi / Di
12
Blazing
b active width of ruling (b ? a)

• Diffracted intensity
• Shift envelope peak to m1
• Blaze condition
• specular reflection off grooves
• also
• ?
• since

Single slit diffraction
Interference pattern
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 gt l
• lmax(m) lB(m1)/m
• Rule-of-thumb
• 40.5 x peak at
• ? (large m)
• Sum over all orders lt 1
• reduction in efficiency with increasing order

2
3
4
5
6
(See Schroeder, Astronomical Optics)
14
Order overlaps
Effective passband in 1st order
Don't forget higher orders!
Intensity
1st order blaze profile
m1
First and second orders overlap!
m2
Passband in 2nd order
Zero order matters for MOS
2nd order blaze profile
Passband in zero order
m0
Wavelength in first order marking position on
detector in dispersion direction (if dispersion
linear)
1st order
0
lU
2lL
2lU
lL
lC
(2nd order)
0
lL
lU
15
Order overlaps
Detector
1st order
2nd order
Zero 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 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)
• grating anomalies not predicted
• 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
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 gt 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

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 gt cDT)

Geometrical factors
Grating parameters
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 10000 okay with wide slit

Immersed gratings
30
Echelle gratings

• Obtain very high R (gt 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
Prism example

• A design for Near-infrared spectrograph of NGST
• DT 8m, c 0.1", D1 D2 86mm, 1 lt l lt 5mm
• R ? 100 required

Raw refractive index data for sapphire
Collimator
Slit plane
Double-pass prismmirror
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 lt 5? obeys scalar theory, little
  polarisation effect (P ? S)
• 5 lt g lt 10? S anomaly at rl ? 2/3 , P peaks at
  lowerrl than S
• 10 lt g lt 18? various S anomalies
• 18 lt g lt 22? anomalies suppressed, S gtgt P at
  large rl
• 22 lt g lt 38? strong S anomaly at P peak, S
  constant at large rl
• g gt 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)
rl
ab