Spectroscopy principlesJeremy Allington-SmithUniversity of Durham

ContentsReflection gratings in low orderSpectral resolutionSlit width issuesGrismsVolume Phase Holographic gratingsImmersionEchellesPrismsPredicting efficiency (semi-empirical)

Generic spectrograph layoutFocal ratios defined asFi = fi / Di

Grating equationInterference condition: path difference between AB and A'B'

Grating equation:

Dispersion:

dxdbf2

"Spectral resolution"Terminology (sometimes vague!) Wavelength resolution dl Resolving powerClassically, in the diffraction limit, Resolving power = total number of rulings x spectral orderI.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 resolutionSpectral resolution:

Projected slit width:Conservation of Etendue (nAW)

Image of slit on detectorCamera focallength

Resolving powerIlluminated grating length:

Spectral resolution (width)

Resolving power:expressed in laboratory terms

expressed in astronomical terms

since andCollimatorfocal ratioPhysicalslitwidthAngular slitwidthTelescopesizeGratinglength

Importance of slit widthWidth of slit determines: Resolving power (R) since Rc = constantThroughput (h)Hence there is always a tradeoffbetween throughput and spectral informationFunction h(c) depends on Point Spread Function (PSF) and profile of extended sourcegenerally h(c) increases slower than c+1 whereas R c-1 so hR maximised at small cSignal/noise also depends on slit widththroughput ( signal)wider slit admits more sky background ( noise)

Signal/noise vs slit widthFor GTC/EMIR in K-band (Balcells et al. 2001)SNR falls as slit includes more sky backgroundOptimum slit width

AnamorphismBeam size in dispersion direction:Beam size in spatial direction:Anamorphic factor:Ratio of magnifications:if b < a, A > 1, beam expandsW increases R increasesimage of slit thinner oversampling worseif b > a, A < 1, beam squashedW reduces R reducesimage of slit wider oversampling betterif b = a, A = 1, beam roundLittrow configurationOutputangleInputangle

Generic spectrograph layoutFi = fi / Di

BlazingDiffracted intensity:

Shift envelope peak to m=1Blaze conditionspecular reflection off grooves: also

since Single slitdiffractionInterferencepatternF = phase difference between adjacent rulingsq = phase difference from centre of one ruling to its edgeb = active width of ruling (b a)

Grating

normal

i

r

Facet

normal

Efficiency vs wavelengthApproximation valid for a > llmax(m) = lB(m=1)/mRule-of-thumb: 40.5% x peak at

(large m)Sum over all orders < 1reduction in efficiency with increasing order

23465(See: Schroeder, Astronomical Optics)

Order overlaps0lU2lL2lUlLWavelength in first order marking position on detector in dispersion direction (if dispersion ~linear)Effective passband in 1st order1st order blaze profilePassband in zero orderm=1m=2m=02nd order blaze profileIntensityPassband in 2nd orderFirst and secondorders overlap!Don't forget higher orders!0lLlU1st order(2nd order)lCZero order matters for MOS

Order overlapsTo eliminate overlap between 1st and 2nd orderLimit 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 spectroscopy1st order2nd orderZeroorderDetector

Predicting efficiencyScalar theory approximateoptical coating has large and unpredictable effectsgrating anomalies not predictedStrong 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 predictedCoating may affect grating properties in complex way for large g (don't scale just by reflectivity!)Two prediction software tools on marketdifferentialintegral

GMOS optical system

Example of performanceGMOS grating setD1 = 100mm, Y = 50DT = 8m, c = 0.5"m = 1, 13.5mm/px Intended to overcoat all with silverDidn't work for those with large groove angle - why?Actual blaze curves differed from scalar theory predictions

GrismsTransmission grating attached to prismAllows in-line optical train:simpler to engineerquasi-Littrow configuration - no variable anamorphismInefficient for r > 600/mm due to groove shadowing and other effects

Grism equationsModified 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 gratingsSo far we have considered surface relief gratingsAn 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 SRPossibility of very large size with high rBlaze condition can be altered (tuned)Encapsulation in flat glass makes more robustDisadvantagesTuning of blaze requires bendable spectrograph!Issues of wavefront errors and cryogenic use

VPH configurationsFringes = planes of constant nBody of grating made from Dichromated Gelatine (DCG) which permanently adopts fringe pattern generated holographically Fringe orientation allows operation in transmission or reflection

VPH equationsModified grating equation: Blaze condition: = Bragg diffractionResolving power:

Tune blaze condition by tilting grating (a)

Collimator-camera angle must also change by 2a mechanical complexity

VPH efficiencyKogelnik's analysis when:

Bragg condition when:Bragg envelopes (efficiency FWHM):in wavelength:

in angle:

Broad blaze requiresthin DCGlarge index amplitudeSuperblaze

VPH 'grism' = vrismRemove bent geometry, allow in-line optical layoutUse prisms to bend input and output beams while generating required Bragg condition

Limits to resolving powerResolving 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 lengthR will increase even if grating overfilled until diffraction-limited regime is entered (l > cDT)GeometricalfactorsGrating parameters

Limits with normal gratingsFor GMOS with c = 0.5", DT = 8m, D1 =100mm, Y =50R and l plotted as function of a A(max) = 1.5 since D2(max) = 150mm R(max) ~ 5000

Simultaneous l rangeNormal SRgratings

Grating

( (mm-1)

( (deg)

( (deg)

(B

RB

A

1200

17.5

0.0

463

3744

B

831

19.7

0.0

757

4396

C

600

8.6

0.0

461

1688

D

600

17.5

0.0

926

3744

H

400

9.7

0.0

764

1918

IA

158

3.6

0.0

720

668

X1

2400

21.0

30.0

444

7043

X2

1800

26.7

30.0

724

9572

X3

1200

17.5

30.0

724

5372

Immersed gratingsBeat the limit using a prism to squash the output beam before it enters the camera: D2 kept small while W can be largePrism is immersed to prism using an optical couplant (similar n to prism and high transmission)

For GMOS R(max) ~ doubled!Potential drawbacks:loss of efficiencyghost imagesbut Lee & Allington-Smith (MNRAS, 312, 57, 2000) show this is not the case

Limits with immersed gratingsFor GMOS with c = 0.5", DT = 8m, D1 = 100mmR and l plotted as function of aWith immersion R ~ 10000 okay with wide slit

Immersedgratings

Grating

( (mm-1)

( (deg)

( (deg)

(B

RB

A

1200

17.5

0.0

463

3744

B

831

19.7

0.0

757

4396

C

600

8.6

0.0

461

1688

D

600

17.5

0.0

926

3744

H

400

9.7

0.0

764

1918

IA

158

3.6

0.0

720

668

X1

2400

21.0

30.0

444

7043

X2

1800

26.7

30.0

724

9572

X3

1200

17.5

30.0

724

5372

Echelle gratingsObtain very high R (> 105) using very long gratingIn Littrow:

Maximising g requires large mr since mrl = 2sing

Instead of increasing r, increase m

Echelle is a coarsegrating with large groove angleR parameter = tang (e.g R2 g = 63.5)Groove angle

Multiple ordersMany orders to cover desired ll: Free spectral range Dl = l/mOrders lie on top of each other:l(m) = l(n) (n/m)Solution:use narrow passband filter to isolate one order at a timecross-disperse to fill detector with many orders at onceCross dispersion may use prisms or low dispersion grating

Echellette example - ESISheinis et al. PASP 114, 851 (2002)

PrismsUseful where only low resolving power is requiredAdvantages:simple - no rulings! (but glass must be of high quality)multiple-order overlap not a problem - only one order!Disadvantages:high resolving power not possibleresolving power/resolution can vary strongly with l

Dispersion for prismsFermat's principle:Dispersion:

Resolving power for prismsBasic definitions:

Conservation of Etendue:

Result:

Comparison of grating and prism: Angular width of resolution elementon detectorAngulardispersionBeam sizeAngularslitwidthDisperser'length''Ruling density'Telescopeaperture

Prism exampleA design for Near-infrared spectrograph* of NGSTDT = 8m, c = 0.1", D1 = D2 = 86mm, 1 < l < 5mmR 100 requiredDouble-pass prism+mirrorCollimatorCameraSlit planeDetector* ESO/LAM/Durham/Astrium et al. for ESARaw refractive index data for sapphire

Prism example (contd)Required prism thickness,t: sapphire: 20mmZnS/ZnSe: 15mmUniformity in dl or R required?For ZnS:

n 2.26 a = 75.3

f = 12.9

Appendix: Semi-empirical efficiency prediction for classical gratings

Efficiency - semi-empiricalEfficiency as a function of rl depends mostly on gDifferent 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 ordersdiscontinuities at (Littrow only) Littrow: symmetry m 1-mOtherwise: no symmetry (rl depends on m,Y) double anomaliesAlso resonance anomalies - harder to predict

m

-3

-2

-1

0

+1

+2

+3

( (a

0.29

0.40

0.67

-

-

0.67

0.40

Efficiency - semi-empirical (contd)Different regimes for blazed (triangular) grooves g < 5obeys scalar theory, little polarisation effect (P S) 5 < g < 10S anomaly at rl 2/3 , P peaks at lowerrl than S10 < g < 18various S anomalies18 < 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: DiffractionGrating Handbook, C. Palmer, Thermo RGL, (www.gratinglab.com) rla=b