Spectroscopy principles Jeremy Allington-Smith University of Durham

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

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

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

"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

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

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

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)

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 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

Generic spectrograph layout Camera Collimator Fi = fi / Di

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

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)

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

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

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

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')

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 = 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 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

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 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)

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 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

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'

* 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

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

Appendix: Semi-empirical efficiency prediction for classical gratings

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 < 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