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## Spectroscopy principles

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Contents

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

Grating equation

n1

n2

A

B

b

a

A’

B’

a

- Interference condition:

path difference between AB and A'B'

- Grating equation:
- Dispersion:

f2

dx

db

"Spectral resolution"

dl

l

- Terminology (sometimes vague!)
- Wavelength resolutiondl
- 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*

Total grating length

Spectral resolution

- Spectral resolution:
- Projected slit width:

Conservation of Etendue(nAW)

Image of slit

on detector

Camera focal

length

Resolving power

Size of spectrograph must scale with telescope size

- 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

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

- Functionh(c) depends on Point Spread Function (PSF) and profile of extended source
- generally h(c) increases slower than c+1whereas 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)

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

dispersion

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

Blazing

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

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

5

6

(See: Schroeder, Astronomical Optics)

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

0

lL

lC

2lL

lU

2lU

(2nd order)

0

lL

lU

Order overlaps

dispersion

Detector

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 2lL = lU)
- Zero order may be a problem in multiobject spectroscopy

Zero

order

2nd order

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 largeg (don't scale just by reflectivity!)
- Two prediction software tools on market
- differential
- integral

GMOS optical system

CCD mosaic

(6144x4608)

Mask field (5.5'x5.5')

Detector (CCD

mosaic)

Science fold

mirror field (7')

Masks and

Integral Field Unit

From

telescope

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

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

Grism equations

- Modified grating equation:
- Undeviated condition:

n'= 1,b = -a = f

- Blaze condition: q=0lB = lU
- Resolving power

(same procedure as for grating)

q = phase difference from centre of one ruling to its edge

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 highr
- 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

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

VPH efficiency

Barden et al. PASP 112, 809 (2000)

- Kogelnik's analysis when:
- Bragg condition when:
- Bragg envelopes (efficiency FWHM):
- in wavelength:
- in angle:
- Broad blaze requires
- thin DCG
- large index amplitude
- Superblaze

VPH 'grism' = vrism

- Remove bent geometry, allow in-line optical layout
- Use prisms to bend input and output beams while generating required Bragg condition

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

Limits with normal gratings

- For GMOS with c= 0.5", DT= 8m,D1 =100mm, Y=50
- Rand l plotted as function of a
- A(max) = 1.5 since

D2(max) = 150mmR(max) ~ 5000

Normal SR

gratings

Simultaneous

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

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

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

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

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

Dispersion for prisms

- Fermat's principle:
- Dispersion:

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'

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

Prism example (contd)

- Required prism thickness,t:
- sapphire: 20mm
- ZnS/ZnSe: 15mm
- Uniformity in dl orR required?
- For ZnS:

n2.26 a=75.3

f= 12.9

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

Efficiency - semi-empirical (contd)

Different regimes for blazed (triangular) grooves

g < 5 obeys scalar theory, little polarisation effect (P S)

5 <g < 10S anomaly at rl 2/3, P peaks at lowerrl than S

10 < g < 18various S anomalies

18 < g < 22anomalies suppressed, S >> P at large rl

22 < g < 38strong S anomaly at P peak, S constant at large rl

g > 38S 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

a=b

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