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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.

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Spectroscopy principles l.jpg

Spectroscopy principles

Jeremy Allington-Smith

University of Durham


Contents l.jpg
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 l.jpg
Generic spectrograph layout

Camera

Collimator

Focal ratios defined as

Fi = fi / Di


Grating equation l.jpg
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 l.jpg
"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 resolution6 l.jpg
Spectral resolution

  • Spectral resolution:

  • Projected slit width:

    Conservation of Etendue(nAW)

     

Image of slit

on detector

Camera focal

length


Resolving power l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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


Generic spectrograph layout11 l.jpg
Generic spectrograph layout

Camera

Collimator

Fi = fi / Di


Blazing l.jpg
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 l.jpg
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 l.jpg
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 overlaps15 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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


Volume phase holographic gratings l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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) = 150mmR(max) ~ 5000

Normal SR

gratings

Simultaneous

l range


Immersed gratings l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
Echellette example - ESI

Sheinis et al. PASP 114, 851 (2002)


Prisms l.jpg
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 l.jpg
Dispersion for prisms

  • Fermat's principle:

  • Dispersion:


Resolving power for prisms l.jpg
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 l.jpg
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 l.jpg
Prism example (contd)

  • Required prism thickness,t:

    • sapphire: 20mm

    • ZnS/ZnSe: 15mm

  • Uniformity in dl orR required?

  • For ZnS:

    n2.26  a=75.3

    f= 12.9



Efficiency semi empirical l.jpg
Efficiency - semi-empirical gratings

  • 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 l.jpg
Efficiency - semi-empirical gratings (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,

(www.gratinglab.com)

rl

a=b


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