1. Interference condition. 2. Dispersion A B. As shown in Figure 1, the path difference between interfering rays AB and A B is a(sin

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1 asic equations or astronomical spectroscopy with a diraction grating Jeremy Allington-Smith, University o Durham, 3 Feb 000 (Copyright Jeremy Allington-Smith, 000). Intererence condition As shown in Figure, the path dierence between interering rays A and A is a(sin α+ sin β) where a is the spacing between the repeated element in the grating rom which relection (or reraction) occurs. The intererence condition is ulilled when the path dierence is equal to multiples, m, o the wavelength o the illuminating light. This gives rise to the grating equation: mρλ = sin α + sin β where ρ = / a is the ruling density, m is the spectral order and λis the wavelength o light.. Dispersion A α β A Figure : Intererence o light by a diraction grating. Although this is considered here as a transmission grating, the same principle applies to a relection grating with suitable care taken with the sign o the angles. a y dierentiating with respect to the output angle we obtain the angular dispersion Detector dλ cos β = dβ mρ The linear dispersion is then dλ dλ dβ cos β = = dx dβ dx mρ χ D T Telescope T Slit s D ψ β α α Grating D W since dβ = dx where is the ocal length o the camera (see Figure ). Figure. Illustration o the principle o a generic grating spectrograph showing the deinition o quantities used in the text. To be consistent with the grating equation, α and β have the same sign i they are on the same side o the grating normal. Jeremy Allington-Smith Page o 6/03/0

2 3. Resolving power In standard textbooks, the resolving power, R = λ λ(where λis the resolution in wavelength), is usually described as being given by the total number o lines in the grating multiplied by the spectral order, hence R * = mρw (sometimes this is called the spectral resolution which can lead to conusion with λ). ut in practice, the resolving power is determined by the width o the image o the slit, s, projected on the detector, s. eore going urther, it is useul to consider the invariance o Etendue in optical systems (note that ibres and some other optical systems do not conserve Etendue but systems made rom normal optics mirrors and lenses - do). This is normally stated as nωa = constant where Ω is the solid angle o radiation incident at a surace o area A in a medium with reractive index n. For our purposes, we may set n = (since we only consider optics in air or vacuum) and consider a one-dimensional analogue: ωa = ω' a' = constant where ω and a are the opening angle o the beam and the aperture dimension respectively. To determine the width o the image o the slit ormed on the detector, we use conservation o Etendue, s θ = s'θ', where the angles at the slit and detector are θ = D and θ = D (Figure 3), so ' s θ slit D D Figure 3: Projection o the slit (let) onto the detector (right). θ' s' Image o slit Jeremy Allington-Smith Page o 6/03/0

3 θ F s ' = s = s θ ' F where the collimator and camera ocal ratios are F = i i D or i =, respectively. i We now express the width o the image o the slit in wavelength units to determine the spectral resolution o the spectrograph dλ dx cos β mρ λ = s' = s = F F sd cos β mρd The length o the intersection between the collimated beam and the plane o the grating (not necessarily the actual physical length o the grating) is so W = D cosβ s λ = mρf W The resolving power, deined as R = λ λ, is then mρλf W R = s Note that this is independent o the details o the camera. This expression is useul or a laboratory experiment since it is expressed in terms o the parameters o the experimental setup: the collimator ocal ratio, F, the grating parameters, ρ and m, the eective grating length, W, and the physical slit width, s. Note that this expression may also be given in terms o the incident and diracted ray angles at the grating by substituting or mρλ rom the grating equation and or the grating length using the previous expression or W. For astronomy, it is more useul to express the resolving power in terms o the angular slit width (projected on the sky), χ, and the telescope aperture diameter, D T. For this we note that and s = χ T Jeremy Allington-Smith Page 3 o 6/03/03

4 D T = = FT = F T D since the spectrograph is directly beam-ed rom the telescope (i.e. no reormatting with ibres is involved). Note that even i the slit is reimaged, the expression below still holds, due to the conservation o Etendue in the reimaging optics. Thus the resolving power is mρλw R = χ D T Note that R R*, the resolving power obtained with a inite slit width is always less than the theoretical maximum which may be obtained with a ininitely narrow slit. This condition is maintained or wavelengths satisying λ < λ * = χd T Thus, long-wavelength applications may approach the theoretical limit; in which case they are said to be diraction-limited. I so, the resolving power is independent o the slit width and it becomes relatively straightorward or the spectrograph to be used with dierent telescopes. For a non-diraction-limited spectrograph, the resolving power obtained would depend on the aperture o the telescope to which it was itted. Note also that the resolving power (when not diraction-limited) is inversely proportional to the telescope aperture diameter. To maintain the same resolving power requires a proportional scaling up in W which implies that the spectrograph should scale in direct proportion to the telescope. 4. Practical example Consider a spectrograph with the ollowing parameters: m =, ρ = 00/ mm, χ = 0.5 arcsec, D = 8m, D = 00mm, λ= 500nm o I the grating tilt is α = 0 then, rom the grating equation, o β = arcsin( m ρλ sin α) = 5. Thus the illuminated grating length is W = D / cosβ = 04mm and the resolving power is R = 560. This compares with the diraction-limited case where R = * Only at wavelengths longer than λ 9 * = µ m will the spectrograph be diraction limited. T Jeremy Allington-Smith Page 4 o 6/03/04

5 Note that, in this example, the angle between the axes o the collimator and camera, o Ψ = α β = 5 (remember the sign convention or these angles), which is probably impractical. 5. Anamorphism The ratio s '/ s is the magniication o the spectrograph in the dispersion direction only. In the direction along the slit the magniication is, in general, dierent. This gives rise to an anamorphic magniication. To see this, we need to consider the shape o the beam exiting rom the grating. I the grating was a mirror (or was used in zero order, m = 0), naturally the output and input beams would have the same circular shape. From Figure, we can seen that the width o the output beam, as seen at the input aperture o the camera, in the direction perpendicular to the slit (the dispersion direction) is D = W cos β but the width in the direction parallel to the slit is D = W cosα Hence the anamorphic actor (Figure 4) is Dispersion direction D D D cos β A = = D cosα y conservation o Etendue, this also means that magniication in the two directions is also dierent. We have already seen that the magniication in the dispersion direction is s' F M λ = = = s F D D but in the along-slit direction, it is simply M x = D Input Output D Figure 4: Anamorphism in the beam shapes. Two examples are shown or the output beam shape. Top: or the normal to collimator coniguration. ottom: or the normal to camera coniguration. Thus, as might expect rom the conservation o Etendue, the ratio o magniication between the two directions is also given by the anamorphic actor since Jeremy Allington-Smith Page 5 o 6/03/05

6 M x D = A M D λ = There are two main conigurations which may be used with blazed gratings, the normal to camera coniguration in which the normal to the grating is pointed generally towards the camera, as illustrated in Figure. In this case β < αso A >. The alternative is the normal to collimator coniguration in which β > α so A <. It is possible to satisy the grating equation or the same wavelength and spectral order in either coniguration but they are not equivalent when their detailed behaviour is examined. To do this, we need to consider the question o blazing the grating. 6. lazing The grating is most eicient when the rays emerge rom the grating as i by direct relection o the acets o which the grating is composed. This is known as the blaze condition. This can be understood by going back to the expression or the intensity rom a diraction grating consisting o N rulings (see e.g. Fundamentals o Optics, Jenkins & White): sin Nφ sin θ I =, sin φ θ λ/b where φ is the phase dierence between the centre o adjacent rulings and θ is the phase dierence between the centre and edge o a single ruling. The second term in the expression is known as the blaze unction. As can be seen rom Figure 5, its eect is to modulate the intererence pattern or a single wavelength. Unortunately, the maximum, when θ = 0, occurs or zero order, m = 0 Intensity -. For practical spectroscopy, we would like to shit the maximum o the blaze unction to occur when, say, m = or some useul wavelength. This can be done by proiling the grating surace so that each periodic unit adopts the shape shown in Figure 6. These acets are tilted at an angle γ to the plane o the grating. The phase dierence between centre and edge o the acet is then λ/a laze unction 0 Spectral order, m Figure 5: Illustrative plot o diracted intensity at a single wavelength versus angle labeled by the spectral order. Jeremy Allington-Smith Page 6 o 6/03/06

7 πcosγ θ = (sin i sin r) ρλ Facet normal Grating normal From Figure 6, it can be seen that i = α γ and r = γ β (Recall the sign convention that α and β have the same sign i they are on the same side o the grating normal.) So the blaze peak condition ( θ = 0 ) occurs when i = r, which is equivalent to simple relection rom the acets, and i Ψ α r γ β α+ β = γ Making use o the identity, x + y x y sin x + sin y = sin cos, we can rewrite the grating equation at the blaze condition as Figure 6: Illustration o blaze condition or relection grating. ρmλ Ψ = sin γ cos since Ψ =α β, where λ is the wavelength or which the blaze condition is satisied and Ψ is the collimator-camera angle which is ixed by the design o the spectrograph (Figure ). Figure 7 shows an idealised blaze unction or a grating blazed at wavelength λ in irst order. It is useul to note (Astronomical Optics, Schroeder) that, in the ideal case, the eiciency o a grating drops to 40.5% o the maximum at wavelengths on either side o the blaze peak o (approximately) Eiciency m= λ + mλ = and λ m mλ = m + and that the useul wavelength range, deined in this way, is λ m= Wavelength Figure 7: Typical relationship between eiciency and wavelength or a blazed diraction grating. Jeremy Allington-Smith Page 7 o 6/03/07

8 λ λ = + λ m or large m. (This is very similar to the expression or the ree spectral range which gives the wavelength range within a single order which may be obtained without being overlapped by light rom dierent orders.) However, real gratings depart rom this idealised situation, especially when dierent polarisation states are considered and when the acet size is comparable with λ. Note also that the behaviour in higher orders is subject to the physical requirement that the sum o the blaze unctions o all orders must not exceed unity. In practice, as indicated in Figure 7, the peak eiciency decreases with increasing order. The actual blaze proile which may be obtained is a complicated matter requiring consideration o Maxwell s equations and is beyond the scope o this paper. For the Littrow coniguration, Ψ = 0 (the incident and diracted rays are parallel), so L mρλ = sin γ So the resolving power in the Littrow coniguration (only) can be expressed as R = D χd T tanγ The relationship between non-littrow and Littrow blaze conditions is λ = cos Ψ L λ L Note that most catalogues o gratings give ρ, γ and λ only, so it is necessary to transorm rom the Littrow case to the actual geometry o the spectrograph. Furthermore the eiciency as a unction o wavelength is usually given or a near-littrow coniguration ( Ψ 0 ). Generally the eiciency o a grating declines slightly with increasing Ψ. 7. Which coniguration is best? We are now in a position to analyse the dierence between the normal-to-camera and normal-to-collimator conigurations noted earlier. It can be seen rom Figure 8, that the two conigurations can both satisy the blaze condition since specular relections are obtained rom the groove acets. In act, these two conigurations are equivalent to using the same grating in positive and negative orders Jeremy Allington-Smith Page 8 o 6/03/08

9 but exchanged end-to-end. To see this, the blaze wavelengths in the + and orders are ound to be + = sin γ cos( Ψ / ) ρ and λ = sin γ cos( Ψ / ) / ρ λ / so λ unless + = λ Ψ π Ψ or γ γ oth transormations are equivalent to turning the grating around end-to-end as illustrated in Figure 8. So we have established that both conigurations are indeed satisactory in terms o satisying the grating equation and the blaze condition. ut they dier in the ollowing respects: The normal-to-camera coniguration has a dilated beam on the grating so the increased value o W will result in a higher spectral resolution. At the same time, the beam collimator camera Ψ grating normal Ruling direction Order m=+ grating normal towards camera eam dilated - higher spectral resolution - larger wavelength range - smaller oversampling red blue collimator Ψ camera grating normal Order m= grating normal towards collimator eam squeezed - lower spectral resolution - smaller wavelength range - larger oversampling red blue grating beam Figure 8: Illustration o the use o the same grating in the normal to camera (top) and normal to collimator (bottom) conigurations. Jeremy Allington-Smith Page 9 o 6/03/09

10 anamorphism will result in a lower magniication in the dispersion direction. Thus the slit will project onto ewer detector pixels (s is smaller). This will reduce the oversampling in the spectrum (generally a bad thing) but also increase the wavelength range that will it on the detector at any one time (a good thing) since the linear dispersion is larger since β is smaller. The normal-to-collimator coniguration has a squashed beam on the grating leading to lower spectral resolution. The magniication is greater in the dispersion direction leading to higher (better) oversampling but a smaller simultaneous wavelength range since the linear dispersion i smaller. Which coniguration is best depends on the details o the spectrograph (or example the slit width and camera speed) and the requirements o the observation to be made. In most case, the normal-to-camera coniguration is to be preerred but this may not always be the case. For this reason, many spectrographs have the capability to use their gratings in either coniguration. However, don t orget to reverse the sense o the grating when changing coniguration, otherwise you could ind yoursel working very ar rom blaze with a consequent large reduction in eiciency this happens quite requently! 8. Grisms A grism is a combination o transmission grating and prism (Figure 9). Naturally, the grating equation applies to this situation but with the modiication that the reractive index o the medium, n, (where we assume that the indices o the γ prism glass and the resin in which the grating is replicated are the φ same, n = n G = n ) must be included: R m ρλ = nsinα + n' sin β Note that the ruling density, ρ, is deined in the plane o the grating, not the plane normal to the optical axis o the spectrograph. For the special case shown in Fig 9 where the input prism ace and the acets are both normal to the optical axis, and where the external medium is air, n '=, the most useul coniguration is whereδ = 0, i.e. the light is undeviated, allowing the camera and collimator to be in line. Here β = α so D β α δ n G n R n' Figure 9: Typical coniguration o a grism. Jeremy Allington-Smith Page 0 o 6/03/0 0

11 mρλ = ( n ) sinφ U Note that, or typical materials ( n =. 5 ), there is roughly a actor 4 dierence in the blaze wavelength or the same grating used in relection or as part o a grism! The advantage o this coniguration is that the monochromatic image o the target will appear at the same location as a direct image obtained with the grism removed. This is also the blaze condition since the phase dierence between the centre and edge o a acet is zero ( θ = 0), since rays emerging rom the centre and edge o a acet pass through identical thickness o glass and are parallel at all times. So we may set, or the special coniguration described, λ = λ. U Working through the same equations as or the relection grating, we ind, as beore, that mρλw R = χ D T Since the grating length is W = D, we can also express the resolving power as cosφ ( n ) tanφd R = χ D T Simple considerations o geometry show that the sizes o the input and output beams in the dispersion direction must be the same. Thus the anamorphic actor is unity. This allows grism-based spectrographs to use a smaller camera than that necessary or a spectrograph employing relection gratings in a non-littrow coniguration. However, it should be noted that the eiciency o grisms with high ruling density, ρ 600/mm, is lower that the equivalent relection grating due to groove shadowing and other eects. It is also useul to consider the case where the acet groove angle,γ, diers rom the prism vertex angle, φ, and where the index o the resin and prism glass are dierent. Then the blaze wavelength is no longer the same as the undeviated wavelength. mρλ n G sinφ + n R cosγ sin γ n arcsin n G R sinφ sinγ I we set γ = φ (as shown in Figure 9), the expression simpliies to: mρλ n G ( n )sinφ cosφ sin φ arcsin sinφ G + nr = mρ( λu + λ* ) nr Jeremy Allington-Smith Page o 6/03/0

12 To take an example actually encountered with LDSS-, a grism with ρ = 600/ mm and o vertex angleφ = 30 (equal to the acet groove angle) was replicated on a glass prism with index n G =. 5. This should have yielded a blaze wavelength λ = λ U = 433nm i the resin and glass indices had been well matched. ut at the irst attempt, a resin with n R =. 60 was used. This caused the blaze to be shited red-wards by λ = 66nm * to 500nm so the grism was remade with a better choice o resin. Acknowledgements Thanks are due to Gordon Love or a critical readthrough. Jeremy Allington-Smith Page o 6/03/0