Dispersive astronomical spectroscopy Jeremy Allington-Smith, University of Durham, 29 July 2002 (Copyright Jeremy Allington-Smith, 2002)

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1 Dispersive astronomical spectroscopy Jeremy Allington-Smith, University of Durham, 9 July 00 (Copyright Jeremy Allington-Smith, 00) I will start by considering the most usual type of dispersing element encountered in astronomical spectroscopy: the ruled plane reflection grating used in low order. Having used this to establish the basic principles, I will consider other types of dispersing element.. Reflection gratings. Interference condition A B α β A B a As shown in Figure, the path difference between interfering rays AB and A B is a(sinα + sin β ) where a is the spacing between the repeated element in the grating from which reflection (or refraction) occurs. The interference condition is fulfilled when the path difference is equal to multiples, m, of the wavelength of 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 λ is the wavelength of light.. Dispersion By differentiating with respect to the output angle we obtain the angular dispersion dλ cos β = dβ mρ χ D T Telescope Figure : Interference of light by a diffraction grating. Although considered here as a transmission grating, the same principle applies to a reflection grating with suitable care taken with the sign of the angles. Jeremy Allington-Smith Page of 9 3/09/0 f T Slit Detector s f D ψ f β α α Grating Figure : Illustration of the principle of a generic grating spectrograph showing the definition of quantities used in the text. To be consistent with the grating equation, α and β have the same sign if they are on the same side of the grating normal. D W

2 The linear dispersion is then dλ dλ dβ cos β = = dx dβ dx mρ f since f dβ = dx where f is the focal length of the camera (see Figure )..3 Resolving power In standard textbooks, the resolving power, R λ δλ (where δλ is the resolution in wavelength), is usually described as being given by the total number of lines in the grating multiplied by the spectral order, hence R * = mρw (sometimes this is called the spectral resolution which can lead to confusion with δλ). But in practice, the resolving power is determined by the width of the image of the slit, s, projected on the detector, s. Before going further, it is useful to consider the invariance of Etendue in optical systems (note that fibres and some other optical systems do not conserve Etendue but systems made from normal optics mirrors and lenses - do). This is normally stated as nωa = constant where Ω is the solid angle of radiation incident at a surface of area A in a medium with refractive 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 of the beam and the aperture dimension respectively. f f s θ slit D D θ' s' Image of slit Figure 3: Projection of the slit (left) onto the detector (right). Jeremy Allington-Smith Page of 9 3/09/0

3 To determine the width of the image of the slit formed on the detector, we use conservation of Etendue, s θ = s'θ ', where the angles at the slit and detector are θ = D f and θ = D f (Fig. 3), so ' θ F s ' = s = s θ ' F f where the collimator and camera focal ratios are F = i i for i =, respectively. D i We now express the width of the image of the slit in wavelength units to determine the spectral resolution of the spectrograph dλ dx cos β mρ f δλ = s' = s = F F sd cos β mρd f The length of the intersection between the collimated beam and the plane of the grating (not necessarily the actual physical length of the grating) is so W = D cos β s δλ = mρf W The resolving power is then R = λ = δλ mρλfw s Note that this is independent of the details of the camera. This expression is useful for a laboratory experiment since it is expressed in terms of the parameters of the experimental setup: the collimator focal ratio, F, the grating parameters, ρ and m, the effective grating length, W, and the physical slit width, s. Note that this expression may also be given in terms of the incident and diffracted ray angles at the grating by substituting for mρλ from the grating equation and for the grating length using the previous expression for W. For astronomy, it is more useful to express the resolving power in terms of the angular slit width (projected on the sky), χ, and the telescope aperture diameter, D T. For this we note that and s = χ f T Jeremy Allington-Smith Page 3 of 9 3/09/0

4 f D f T = = FT = F T D since the spectrograph is directly beam-fed from the telescope (i.e. no reformatting with fibres is involved). Note that even if the slit is reimaged, the expression below still holds, due to the conservation of Etendue in the reimaging optics. Thus the resolving power is R = mρλw χ D T Note that R R * : the resolving power obtained with a finite slit width is always less than the theoretical maximum which may be obtained with a infinitely narrow slit. This condition is maintained for wavelengths satisfying λ < λ * = χd T Thus, long-wavelength applications may approach the theoretical limit; in which case they are said to be diffraction-limited. If so, the resolving power is independent of the slit width and it becomes relatively straightforward for the spectrograph to be used with different telescopes. For a non-diffraction-limited spectrograph, the resolving power obtained would depend on the aperture of the telescope to which it was fitted. Note also that the resolving power (when not diffraction-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 following parameters: m =, ρ = 00/ mm, χ = 0.5 arcsec, D = 8m, D = 00mm, λ = 500nm! If the grating tilt is α = 0 then, from the grating equation,! β = arcsin( m ρλ sinα) = 5. Thus the illuminated grating length is W = D / cos β = 04mm and the resolving power is R = 560. This compares with the diffraction-limited case where R * = Only at wavelengths longer than λ * = 9µ m will the spectrograph be diffraction limited. Note that, in this example, the angle between the axes of the collimator and camera,! Ψ = α β = 5 (remember the sign convention for these angles), which is probably impractical. T Jeremy Allington-Smith Page 4 of 9 3/09/0

5 .5 Anamorphism The ratio s '/ s is the magnification of the spectrograph in the dispersion direction only. In the direction along the slit the magnification is, in general, different. This gives rise to an anamorphic magnification. To see this, we need to consider the shape of the beam exiting from the grating. If 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 Fig., we can seen that the width of the output beam, as seen at the input aperture of the camera, in the direction perpendicular to the slit (the dispersion direction) is D = W cos β but the width of the beam in the direction parallel to the slit is just D = W cosα Hence the anamorphic factor (Fig. 4) is A = D D = cos β cosα D By conservation of Etendue, this also means that magnification in the two directions is also different. We have already seen that the magnification in the dispersion direction is D D s' F M λ = = = s F f D f D D but in the along-slit direction, it is simply f M x = f Thus, as might expect from the conservation of Etendue, the ratio of magnification between the two directions is also given by the anamorphic factor since M x D = A M D λ = Input Output Figure 4: Anamorphism in the beam shapes. Two examples are shown for the output beam shape. Top: for the normal to collimator configuration. Bottom: for the normal to camera configuration. Dispersion is vertical. Jeremy Allington-Smith Page 5 of 9 3/09/0

6 There are two main configurations which may be used with blazed gratings, the normal to camera configuration in which the normal to the grating is pointed generally towards the camera, as illustrated in Figure. In this case β < α so A >. The alternative is the normal to collimator configuration in which β > α so A <. It is possible to satisfy the grating equation for the same wavelength and spectral order in either configuration but they are not equivalent when their detailed behaviour is examined. To do this, we need to consider the question of blazing the grating..6 Blazing The grating is most efficient when the rays emerge from the grating as if by direct reflection off the facets of which the grating is composed. This is known as the blaze condition. This can be understood by going back to the expression for the intensity from a diffraction grating consisting of N rulings (e.g. Jenkins & White 976): = sin Nφ sin θ I, sin φ θ where φ is the phase difference between the centre of adjacent rulings andθ is the phase difference between the centre and edge of a single ruling. The second term in the expression is known as the blaze function. As can be seen from Fig. 5, its effect is to modulate the interference pattern for a single wavelength. Unfortunately, the maximum, when θ = 0, occurs for zero order, m = 0. For practical spectroscopy, we would like to shift the maximum of the blaze function to occur when, say, m =, for some useful wavelength. This can be done by profiling the grating surface so that each periodic unit adopts the shape shown in Fig. 6. λ/b These facets are tilted at Blaze function an angle γ to the plane of the grating. The phase difference between centre and edge of the facet is then λ/a Intensity - 0 Spectral order, m Figure 5: Illustrative plot of diffracted intensity at a single wavelength versus angle labeled by the spectral order. Jeremy Allington-Smith Page 6 of 9 3/09/0

7 π cosγ θ = (sin i sin r) ρλ Facet normal Grating normal From Fig. 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 if they are on the same side of the grating normal.) So the blaze peak condition (θ = 0) occurs when i = r, which is equivalent to simple reflection from the facets, and i Ψ α r γ β α + β = γ Making use of 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 of blaze condition for reflection grating. ρmλ B Ψ = sinγ cos since Ψ = α β, where λb is the wavelength for which the blaze condition is satisfied and Ψ is the collimator-camera angle which is fixed by the design of the spectrograph (Fig. ). Fig. 7 shows an idealised blaze function for a grating blazed at wavelength λb in first order. It is useful to note (Schroeder 000) that, in the ideal case, the efficiency of a grating drops to 40.5% of the maximum at wavelengths on either side of the blaze peak at (approximately) Efficiency m= λ + mλb m and λ = mλb = m + m= where λ B is defined at the order of interest (i.e. λ B (m=)/m) and that the useful wavelength range, defined in this way, is λ B Wavelength Figure 7: Typical relationship between efficiency and wavelength for a blazed diffraction grating. Jeremy Allington-Smith Page 7 of 9 3/09/0

8 λ λ = + λ B m for large m. (This is very similar to the expression for the free spectral range which gives the wavelength range within a single order which may be obtained without being overlapped by light from different orders.) However, real gratings depart from this idealised situation, especially when different polarisation states are considered and when the facet size is comparable with λ. Note also that the behaviour in higher orders is subject to the physical requirement that the sum of the blaze functions of all orders must not exceed unity. In practice, as indicated in Fig. 8, the peak efficiency decreases with increasing order. The actual blaze profile which may be obtained is a complicated matter requiring consideration of Maxwell s equations and is beyond the scope of this paper. For the Littrow configuration, Ψ = 0 (the incident and diffracted rays are parallel), so L mρλ = sinγ B ISACgratingefiency(fromESOETC) di li id Wlh() Figure 8: Efficiency of ISAAC grating in different orders. So the resolving power in the Littrow configuration (only) can be expressed as Jeremy Allington-Smith Page 8 of 9 3/09/0

9 R = D χd T tanγ The relationship between non-littrow and Littrow blaze conditions is λ = cos( Ψ ) L B λb Note that most catalogues of gratings give ρ, γ and λ only, so it is necessary to transform from the Littrow case to the actual geometry of the spectrograph. Furthermore the efficiency as a function of wavelength is usually given for a near-littrow configuration ( Ψ 0 ). Generally the efficiency of a grating declines slightly with increasing Ψ..7 Which configuration is best? We are now in a position to analyse the difference between the normal-to-camera and normal-to-collimator configurations noted earlier. collimator camera Ψ grating normal Ruling direction Order m=+ grating normal towards camera Beam dilated - higher spectral resolution - larger wavelength range - smaller oversampling red blue collimator Ψ camera grating normal Order m= grating normal towards collimator Beam squeezed - lower spectral resolution - smaller wavelength range - larger oversampling red blue grating beam Figure 9: Illustration of the use of the same grating in the normal to camera (top) and normal to collimator (bottom) configurations. Jeremy Allington-Smith Page 9 of 9 3/09/0

10 It can be seen from Fig. 9, that the two configurations can both satisfy the blaze condition since specular reflections are obtained from the groove facets. In fact, these two configurations are equivalent to using the same grating in positive and negative orders but exchanged end-to-end. To see this, the blaze wavelengths in the + and orders are found to be + sinγ cos( ) = Ψ B ρ and λ γ ( ) sin cos Ψ B = / ρ λ / + so λ B = λb unless γ γ which is equivalent to turning the grating around end-to-end as illustrated in Fig. 9. So we have established that both configurations are indeed satisfactory in terms of satisfying the grating equation and the blaze condition. But they differ in the following respects: The normal-to-camera configuration has a dilated beam on the grating so the increased value of W will result in a higher spectral resolution. At the same time, the beam anamorphism will result in a lower magnification in the dispersion direction. Thus the slit will project onto fewer 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 fit on the detector at any one time (a good thing) since the linear dispersion is larger since β is smaller. The normal-to-collimator configuration has a squashed beam on the grating leading to lower spectral resolution. The magnification is greater in the dispersion direction leading to higher (better) oversampling but a smaller simultaneous wavelength range since the linear dispersion if smaller. Which configuration is best depends on the details of the spectrograph (for example the slit width and camera speed) and the requirements of the observation to be made. In most case, the normal-to-camera configuration is to be preferred but this may not always be the case. For this reason, many spectrographs have the capability to use their gratings in either configuration. However, don t forget to reverse the sense of the grating when changing configuration, otherwise you could find yourself working very far from blaze with a consequent large reduction in efficiency this happens quite frequently!.8 Example of a modern high-performance spectrograph. An example of gratings employed in a high-performance grating spectrograph is shown in Fig. 0 (refer to Table for details). For some, a proprietary silver coating was used instead of the usual aluminium to improve the efficiency. It was intended to silver-coat all the gratings but in some cases the performance was unexpectedly poorer than with an aluminium coating. These were all in cases where the groove angle was large. Jeremy Allington-Smith Page 0 of 9 3/09/0

11 Absolute efficiency of fitted GMOS gratings (Littrow, unpolarised) Efficiency (%) Wavelength (microns) B00 R83 B600 R600 R400 R50 Table : Details of GMOS gratings. λ B and R B are the wavelength and resolving power (0.5 arcsec slitwidth) at blaze. GMOS grating name Coating Ruling density, ρ (mm - ) Groove angle, γ (deg) Nominal Actual λ B (nm) R B λ B (nm) Dispersion, dλ/dx (nm/pixel) B00 Al R83 Al B600 Ag R600 Al R400 Ag R50 Ag Grisms Figure 0: The efficiency of the gratings used in the Gemini Multiple Object Spectrographs (GMOS) from measurements by the vendor in Littrow configuration. A grism is a combination of transmission grating and prism (Fig. ). Naturally, the grating equation applies to this situation but with the modification that the refractive index of the medium, n, (where we assume that the indices of the prism glass and the resin in which the grating is replicated are the same, n = n G = n ) must be included: R Jeremy Allington-Smith Page of 9 3/09/0

12 m ρλ = nsinα + n' sin β Note that the ruling density, ρ, is defined in the plane of the grating, not the plane normal to the optical axis of the spectrograph. For the special case shown in Fig. where the input prism face and the facets are both normal to the optical axis, and where the external medium is air, n '=, the most useful configuration is whereδ = 0, i.e. the light is undeviated, allowing the camera and collimator to be in line. Here β = α = φ so mρλ = ( n ) sinφ U Note that, for typical materials ( n =. 5 ), there is roughly a factor 4 difference in the blaze wavelength for the same grating used in reflection (Littrow) or as part of a grism! The advantage of this configuration is that the monochromatic image of 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 difference between the centre and edge of a facet is zero ( θ = 0 ), since rays emerging from the centre and edge of a facet pass through identical thickness of glass and are parallel at all times. So we may set, for the special configuration described, λ = λ. B U Working through the same equations as for the reflection grating, we find, as before, that mρλw R = χ D T φ γ Since the grating length is D W =, we can also express cosφ the resolving power as D α β δ ( n ) tanφ D R = χ D T Simple considerations of geometry show that the sizes of the input and output beams in the dispersion direction must be the same. Thus the anamorphic factor is unity. This allows grism-based spectrographs to use a smaller camera than that necessary for a n G n R n' Figure : Typical configuration of a grism. Jeremy Allington-Smith Page of 9 3/09/0

13 spectrograph employing reflection gratings in a non-littrow configuration. However, it should be noted that the efficiency of grisms with high ruling density, ρ 600/mm, is lower that the equivalent reflection grating due to groove shadowing and other effects. It is also useful to consider the case where the facet groove angle, γ, differs from the prism vertex angle, φ, and where the index of the resin and prism glass are different. Then the blaze wavelength is no longer the same as the undeviated wavelength. mρλ B n G sinφ + n R n cosγ sin γ arcsin n G R sinφ sinγ If we set γ = φ (as shown in Fig. ), the expression simplifies to: mρλ B n G ( n )sinφ cosφ sin φ arcsin sinφ G + nr = mρ( λu + λ* ) nr To take an example actually encountered with LDSS-, a grism with ρ = 600/ mm and! vertex angleφ = 30 (equal to the facet groove angle) was replicated on a glass prism with index n G =. 5. This should have yielded a blaze wavelength λ B = λ U = 433nm if the resin and glass indices had been well matched. But at the first attempt, a resin with n R =. 60 was used. This caused the blaze to be shifted red-wards by λ* = 66nm to 500nm so the grism was remade with a better choice of resin. 3. Volume Phase Holographic (VPH) Gratings [This section is drawn largely from Barden et al. 000]. 3. Basic principles The dispersing elements discussed so far generate the interference between beams by the microstructure of the surface (surface relief SR gratings). There is also a class of dispersive element that modulates the refractive index within the volume of the material of the grating thus causing phase differences between rays passing through adjacent parts of the material: volume phase holographic (VPH) gratings. This should not be confused with holographic gratings which are SR gratings where the surface shape is generated by a holographic photo-etching process. In VPH gratings, the refractive index is modulated harmonically as a function of distance within a substrate. If the modulation is sinusoidal, the refractive index is given by n [ πρ ( xsinγ cosγ )] ( x, z) = n + n cos z g g g g + Jeremy Allington-Smith Page 3 of 9 3/09/0

14 Where γ is the angle between lines of constant n g (the fringes ) and the grating surface. z and x are coordinate directions parallel to the optical axis and the orthogonal direction in which dispersion occurs. The geometry can be varied so that the fringes are perpendicular, parallel or at some arbitrary angle to the surfaces of the grating (Fig. ). The body of the grating is dichromated gelatine (DCG), whose refractive index is permanently modified by illumination with UV light in a holographic arrangement. The DCG is generally sandwiched between sheets of optical-quality glass which also protect it from the environment. VPH gratings obey the grating equation mρλ = sin α + sin β where ρ is the frequency of the intersection of the grating fringes with the grating surface. Thus ρ = ρ g sinγ where ρ g is the true fringe density so ρ = ρ g for fringes perpendicular to the grating surface since γ = π/. Here the incident and output angles are measured in air. The diffracted energy pattern is governed by Bragg diffraction giving a maximum when γ = π/ α α β γ α+π γ α Figure : Basic geometry of Volume Phase Holographic gratings showing the Bragg conditions for two different geometries. For simplicity the modification of the ray angles in the VPH medium is not shown. Jeremy Allington-Smith Page 4 of 9 3/09/0

15 Figure 3: Efficiency of a VPH grating with ρ = 00/mm (= ν) in unpolarized light, inclusive of substrate material and surface reflection losses. The superblaze envelope is indicated. From Barden et al. (000). mρλ = B n g sinα g where α g is the angle of incidence inside the grating medium with respect to the fringe plane. For the special case for fringes perpendicular to the grating surface when the incident angle is measured in air, we can use Snell s law to write n g sin α g = sinα The expression for resolving power can be obtained by noting that the illuminated length of grating is W = D /cosα, where D is the diameter of the beam from the collimator, so R = mρλw χd T = mρλ D χd cosα T As with all gratings, the performance is difficult to model, but Kogelnik s analysis (see Barden et al.) is applicable in the case πλdρ n g g. > 0 where d is the thickness of the phase-modulating layer (i.e. the DCG). In this case, the Bragg condition is satisfied when Jeremy Allington-Smith Page 5 of 9 3/09/0

16 n g λ d. The width of the blaze peak in this approximation is given at FWHM in terms of angle of angular and spectral Bragg envelopes. These are, respectively, the range of incidence angle, α, at constant wavelength and the range of wavelength, λ, at fixed incidence angle: α ρ g d and λ λ n g = ρ g α g ρ g α tan tan g d Fig. 3 shows spectral Bragg envelopes for different incident angles and illustrates the superblaze envelope which delineates the maximum efficiency which may be obtained by choosing the optimum combination of wavelength and incidence angle. Figure 4: Changes in the optical path to the camera as the Bragg condition is changed by tilting the VPH grating. 3. Using VPH gratings The advantage of VPH gratings is that near to the Bragg peak the efficiency can be very high, for example 88% in Fig. 4 at the superblaze peak, compared with SR gratings which rarely exceed 80%. However there are two features to be taken into account. The blaze peak is relatively narrow making it of limited use in applications where a wide simultaneous wavelength range is required. Making the peak broader requires a reduction in the DCG thickness, d, or an increase in n g which is limited by technological considerations. Fortunately, unlike the case for SR gratings, the blaze (=Bragg) wavelength may be altered... Changing the blaze condition involves changing not only the tilt of the grating but also the angle between the collimator and camera. This is because the incident and output angles must be equal and it is by varying the incident angle that the Bragg condition is modified. This imposes the geometry shown in Fig. 4 and imposes significant mechanical complexity into the spectrograph. Of course the angle of the fringes may be altered instead but this cannot be done without remaking the grating and so is directly analogous to altering the angle of the grooves in an SR grating. Jeremy Allington-Smith Page 6 of 9 3/09/0

17 n n g n α δ D φ Figure 5: Example of a vrism - a VPH grism sandwiched between two equal prisms to allow an in-line geometry Potential advantages include not only the higher peak efficiency but also the possibility of making very high density gratings in very large sizes, something which is difficult for SR gratings. 3.3 VPH grisms To try and avoid the bent geometry required for a VPH grating, one can combine the grating with a prism to make a vrism, analogous to a grism. There are various ways to do this and although it is possible to make a vrism using a prism cemented to only one side of the VPH grating, especially if slanted fringes are used (e.g. LDSS++; Glazebrook 998), the example of the use of two equal prisms (index n ) is shown in Fig. 5. The prism bends the light to the correct Bragg angle before it strikes the grating and reverses the bend after it, so that the input and output beams are in-line. It is straightforward to show that the resolving power in the Bragg condition is given by mρλw mρλ R = = D ( + tanδ tanϕ) χd χd T T where the angles are defined in the diagram. In this configuration, α = φ and the refracted angle in the prism is given by sinϕ δ = ϕ arcsin n Jeremy Allington-Smith Page 7 of 9 3/09/0

18 4. Limits to resolving power immersed gratings. [This section is drawn largely from Lee and Allington-Smith, 000.] The expression for resolving power makes it plain that increasing the ruling density of the grating (ρ), or the order of diffraction (m), will increase the resolving power at constant wavelength. But the illuminated grating length, W, (or more precisely, the length of the intersection of the beam with the grating plane) also has a dominant effect. In fact, this is what puts limits on the resolving power which can be obtained. Referring to Fig., it can be seen that increasing W will at some point produce a beam so wide (D ) that it overfills the camera. If this happens, the resolving power will continue to increase (at least until the number of rulings illuminated becomes small enough to enter the diffraction limited Figure 6: Predicted resolving power for a set of gratings considered for GMOS. These are labelled A, B, C, D, H and IA. Also shown are predictions for immersed gratings, X, X and X3, discussed in the text. Jeremy Allington-Smith Page 8 of 9 3/09/0

19 regime), but the throughput will decrease. This is usually unacceptable. The problem is illustrated in Fig. 6, where for the example of GMOS, the attainable resolving power for 0.5 arcsec slitwidth has been calculated by tilting the gratings considered to the angle required to place the desired wavelength at the centre of the field. The simultaneous wavelength range (i.e. that which fits on the detector all at once if the slit is centrallyplaced) is indicated by the width of each trace, but the important thing to note is that the maximum achievable resolving power is ~5000. This is where the beam starts to overfill the camera. The wavelength for which each grating reaches this limits depends on the ruling density (and order), but the maximum resolving power is the same. This can be seen by rewriting the resolving power in terms of the input and output angles as R = mρλw χd D = χ T D T (sinα sin β ) cos β + substituting from the grating equation and noting that W = D /cosα.. The expression is now independent of the grating ruling and spectroscopic order increasing ρ or m will not increase the resolving power above the geometrical limit. However this limit can be overcome using a prism attached to the grating surface. This process is called immersion since there is an optical couplant (an index-matching liquid, gel or cement) filling the gap between the grating surface (which may be structured in the case of a surface-relief grating) and the output face of the prism. Fig. 7 shows an example with rays traced through from the collimator to the camera. The important point to note is that the illuminated length of grating has been increased without making the diameter of the beam entering the camera too large. The trick is accomplished simply by using the anamorphic ability of the prism coupled to the grating the beam is squeezed by the prism before it reaches the camera. In this example, the beam is almost unaffected Table : Details of unimmersed and immersed gratings discussed in the text. Grating ρ (mm - ) γ (deg) φ (deg) λ B R B A B C D H IA X X X Jeremy Allington-Smith Page 9 of 9 3/09/0

20 camera prism collimator grating Immersed 800/mm grating with 35 deg prism, nm Figure 7: Example of ray-tracing through an immersed grating. by the prism before hitting the grating because the input face of the prism is almost perpendicular to then incoming light. To calculate the effect is not straightforward as there are a lot of angles to be calculated. However Fig.6 shows the results of calculations for three example immersed configurations as detailed in Table. φ is the vertex angle of the prism and λ B and R B are the nominal wavelength and resolving power at blaze. In summary, it can be seen that the maximum resolving power is now about twice what it was! The main points are: Immersed gratings allow the maximum resolving power to be roughly doubled. Higher density gratings may be used than without immersion which, together with the increased illuminated length of grating, creates the necessary conditions for this to happen The dispersion takes place in a medium (index n) so the grating must be designed accordingly using the modified grating equation: mρλ = n(sinα + sin β ) Jeremy Allington-Smith Page 0 of 9 3/09/0

21 γ D γ W Figure 8: Basic layout of an Echelle grating used in near- Littrow configuration. In practice this means that the grating works at a wavelength n times higher than in the unimmersed case. This must be taken into account when specifying the grating to be immersed. Note especially that the blaze condition will be satisfied at longer wavelength for a given groove angle. Because of the extra glass surfaces, there is the possibility of a reduction in throughput and ghost reflections. However this effect can be small see Lee & Allington-Smith 000 making immersed gratings a very good way to greatly improve spectrograph performance. 5. Limits to resolving power echelles Another option to increase the resolving power is to use a very coarse grating in very high order. As already discussed, this will not increase the resolving power by itself unless the illuminated grating length is made very large. One configuration which allows this is the echelle format. The resolving power in the near-littrow configuration in which it is usually operated is mρλw R = = χd D χ T D T tanγ as derived in Section.6, where γ is the groove angle. (Echelle gratings are often characterised by an R-parameter equal to tanγ so R- implies γ = 63.5 deg). In the configuration shown in Fig. 8, the illuminated grating length can be very large so R becomes large. However, this means that γ is large so mρλ = sinγ must also be large. There are two strategies for maximizing the product mρ. One is to increase ρ while Jeremy Allington-Smith Page of 9 3/09/0

22 Cross-dispersion Primary dispersion 300 Figure 9: Illustrative cross-dispersed spectrogram showing a simplified layout on the detector. The numbers 0-6 label the different spectral orders. The numbers labelling the vertical axis are the wavelength (nm)at the lowest end of each complete order. Other wavelengths are labelled for clarity. For simplicity the orders are shown evenly spaced in cross-dispersion. keeping the spectral order low this helps to maximise free spectral range ( λ λ/m), or to keep ρ small but increase m. The latter option is made desirable because of the difficulty in engineering a large grating with high ruling density but comes at the price of having to deal with many different orders which lie on top of each other on the detector. Each order will differ only slightly in wavelength at the same location on the detector λ(m) = λ()/m so, for example λ() = 550nm will occur at the same location as λ() = 55nm, making it difficult to observe simultaneously over a significant wavelength range even if sufficiently precise passband filters are available. One solution to this problem is to cross-disperse the spectra so that the different orders are separated on the detector surface in a direction perpendicular to the primary axis of dispersion. With a suitable choice of design parameters, one order will roughly fill the full extent of the detector in the primary dispersion direction. The number of orders which will fit on the detector is determined by the amount of cross-dispersion. With Jeremy Allington-Smith Page of 9 3/09/0

23 α φ β L D D t Prism, index n Figure 0: Rays traced through a prism at one wavelength. suitable choices of design parameters it is possible to cover a wide wavelength range, say from nm, as shown in Fig. 9, in a single exposure without gaps between orders. 6. Prisms The significance of prisms in modern astronomy is that multi-object spectrographs often require only low spectral resolution, and, if so, prisms can be used with the big advantage over gratings that they do not produce multiple orders that have the potential to contaminate the spectra of other objects. Consider rays from a collimator traversing a prism with a refractive index n(λ) as shown in Fig. 0. The prism is defined by its vertex angle φ and the beam by its diameter D.The refracted rays are then focussed by a camera. From Fermat s principle, the upper and lower rays have paths related by tn = L cosα The dispersion is d λ = dβ dλ dn dn dβ From the diagram β = π φ α Jeremy Allington-Smith Page 3 of 9 3/09/0

24 so dβ dα = and since dn dn dn Lsinα D = = dα t t we can write dn dn D = = dβ dα t so the dispersion is d D = dβ t λ dλ dn The resolving power is therefore λ dλ R where δλ = δβ δλ d β Where δβ is the projected width of the spectral resolution element. Conservation of Etendue allows us to relate this to the angular slit width projected on the sky, χ, χ = δβ D T D where D T is the telescope aperture and D is the aperture of the camera. Thus the width of the resolution element is D dλ χdt δλ = = t dn D χd t T dλ dn since D = D. Hence R = λ χd T dn t = dλ λ χd T R * prism where, by analogy with the expression for the resolving power of a grating, we define the diffraction-limited resolving power of prism as * dn R prism = t dλ The equivalent for a grating is Jeremy Allington-Smith Page 4 of 9 3/09/0

25 Resolving power R Wavelength (µm) R for t=0mm ZnSe R for t=0mm ZnS R for t=5mm Sappphire Fit for ZnS Fit for sapphire Figure : Calculated resolving power for the prism examples discussed in the text for the dimensions shown in the key. The smooth lines are polynomial fits to the datapoints derived from glass vendors low-precision data. * R grat = mρw where the symbols are defined above. Thus the equivalent to the size of the intersection of the beam with the grating surface for a prism is the width of its base and the equivalent of the ruling density is the derivative of refractive index with wavelength. A practical example is provided by a design explored for a wide-field infrared spectrograph for the Next Generation Telescope (originally D T = 8m). Here a prism is used to provide a resolving power of R!00 for a slitwidth, χ = 0. arcsec λ dn The required size of the prism, t, was calculated from R = t making use of χdt dλ refractive index data for two candidate materials, sapphire and ZnS/ZnSe, which have good properties in the -5µm wavelength range required. The raw data were the refractive index of the material. From this it was discovered that thicknesses of t =0mm for ZnS/ZnSe or t = 5mm for sapphire would be sufficient. For the example of ZnS, the tilt angle was calculated from L = sinα giving D tn = L cosα with Jeremy Allington-Smith Page 5 of 9 3/09/0

26 D α = arctan nt where the incident beam size is fixed by the spectrograph design to be D = 86mm. Thus we find α = 75.3 deg since n.6 and t = 0mm has already been calculated. The prism vertex angle is found from sin φ = t L = t D sinα to be φ =.9 deg. A spreadsheet calculation was used to derive the variation of resolving power with wavelength as shown in Fig.. It can be seen that ZnS gives the least variation over the wavelength range but the non-linearity in dispersion gives a large variation in R. References Barden, S., Arns, J., Colburn, W. and Williams, J., 000. PASP,, 809 Glazebrook, K., 998. AAO Newsletter, 87,. Jenkins, F. and White, H., 976. Fundamentals of Optics, 4 th Edn, McGraw Hill. Lee, D. and Allington-Smith, J.R., 000. MNRAS, 3, 57. Schroeder, D., 000. Astronomical Optics, nd edn, Academic Press. Jeremy Allington-Smith Page 6 of 9 3/09/0

27 Appendix A: Semi-empirical estimation of grating efficiency. [Based partly on: Diffraction Grating Handbook, C. Palmer, Thermo RGL, ( Consideration of the following points help us to predict the qualitative behaviour of the curve of efficiency vs wavelength. The efficiency as a function of ρλ depends mostly on γ. Different behaviour is seen depending on polarisation: P - parallel to grooves (TE) S - perpendicular to grooves (TM) Overall, the peak efficiency occurs at ρλ = sinγ for Littrow examples. Passoff anomalies occur when light is diffracted from an order at an angle of β = π/. Since the light cannot propagate, the energy is redistributed into other orders causing discontinuities. In the Littrow case, these occur at ρλ a = m i.e. at the following values: m ρ λ a In the Littrow configuration, there is a symmetry, m -m, but otherwise there is no symmetry since ρλ depends on both m and Ψ, leading to double the anomalies found in the Littrow case. There are also resonance anomalies, but these are harder to predict Armed with this information, we can identify different regimes for gratings with blazed (triangular) grooves: γ < 5 : obeys scalar theory, little polarisation effect (P S) 5 < γ < 0 : S anomaly at ρλ /3, P peaks at lower ρλ than S 0 < γ < 8 : various S anomalies 8 < γ < : anomalies suppressed, S >> P at large ρλ < γ < 38 : strong S anomaly at P peak, S constant at large ρλ γ > 38 : S and P peaks very different, efficient in Littrow only The following graphs from the Diffraction Grating Handbook illustrate this behaviour. Jeremy Allington-Smith Page 7 of 9 3/09/0

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