Spectrographs C A Griffith, Class Notes, PTYS 521, 2016 Not for distribution
1 Spectrographs and their characteristics A spectrograph is an instrument that disperses light into a frequency spectrum, which is generally recorded with the use of a camera that focuses the dispersed light on a detector 11 Diffraction Gratings & Dispersion Elements The principal method for dispersing light in UV to IR wavelengths is through diffraction gratings In the same way a large series of equally spaced slits can produce a very sharp diffraction pattern, a reflective diffraction grating, a mirror cut with many equally spaced grooves at similar angles, can also cause a diffraction pattern Transmission diffractive gratings, such that light passes through a plate, are also used The plates often contain hundreds to thousands of grooves per millimeter, such that the originals are cut by diamond across a metal surface Replica casts can be made from liquid plastic, which when cooled can be coated with a reflective material Light can also be dispersed by prisms and grisms Prisms are used for low dispersion needs, such as cross-dispersers To achieve high precision, prisms mustbelarge,andwhichthenleadstogreaterabsorption and lower throughput In addition, prisms do not disperse uniformly from blue to red A grism (or grating prism) combines a prism and grating so that light at a chosen central wavelength passes through The advantage grisms is that the same camera can be used both for imaging (without the grism) and spectroscopy (with the grism) without having to be moved Grisms are inserted into a camera beam that is already collimated They create a dispersed spectrum centered on the object s location in the camera s field of view Interferometry is another method for measuring spectra We ll discuss that later The geometry of a reflective blazed grating is shown in Fig 1 Light from adjacent grooves travel light path distances that depend on the angle of the incident and reflected diffracted light(α and β) as well as the spacing of the groves, d Constructive interference occurs for a wavelength λ obeys the grating equation: d(sin α + sin β) = mλ 1 Fig 1 A grating with a blaze angle θ (a) Consider a wavefront incident on a grating at an angle α The portion of the wavefront reaching groove B must travel an extra distance d sin(α) (b) Light reflected at an angle β must travel an extra distance d sin(β) Here m is the number of wavelengths of the path difference, and is called the spectral order Note then that multiple spectra are produced, one for each order Also note that light of wavelength λ in the first spectral order (m=1) interferes constructively in the same direction (same value of β, Fig 1) as light of wavelength λ/2 in the second spectral order (m=2) Filters can be used to select orders, and thus effectively wavelength regions, and omit others Also note that the zeroth order diffraction (m=0) has no wavelength dependence, and therefore sends all the light in the sin(β) = - sin(α) direction (Figs 2,3) The grating is blazed in order to avoid diffracting the order of interest in the zeroth order diffraction direction θ 2
Fig 2 Here we can see that the 0-order diffracted light occurs when β = -α Note that these angles are defined in terms of the normal to the grating (not facet) and are opposite signs at opposite sides to the normal This light is white and best scattered away from the desired the order to be measured Another aspect of diffraction gratings is that when the relationship between the incident light and the mth-order diffracted light describes mirror reflection with respect to the facet surface of the grooves, most of the energy is concentrated into this mth-order diffracted light (Fig 3) The facet angle of the grooves at this point is called the blaze angle, θ, satisfies the following: α+β = 2θ 2 Note that the blaze angle depends on the incident and reflected angles Alternatively, based on Eq 1, it depends on the incident angle and the wavelength We can combine equation 1 and 2, and determine the wavelength at which the maximum intensity of diffracted light is concentrated for a given incident angle, order, and blaze angle Combining Eqs 1 and 2, the blaze wavelength is: λ B = 2d m sin(θ)cos(α θ) The efficiency of the blazed grating in order, m, illuminated at an incident angle, α, peaks at this wavelength The blaze angle is set so as to avoid mixing the spectrum with the zero order light (eg as in Fig 3) Fig 3 In this configuration the grating is illuminated with incident normal light The blaze wavelength is that for which light is scattered like a mirror reflection off of the groove facet, as shown This light is scattered away from the 0th order light, as desired The angular dispersion, β/ λ, the change in the dispersion angle that results from a change in the wavelength, provides a measure of the spectral resolution possible for a certain slit width This value can be derived from the grating equation, written as sin β = mλ/d - sin α: A = β λ = β sinβ sinβ λ β λ = 1 sinβ sinβ/ β λ β λ = 1 m cosβ d = sin(α)+sin(β) λcos(β) Thus, the angular dispersion can be increased by going to a higher spectral order, m, or by using a spectrograph with a narrower groove spacing, d 12 Getting light to the spectrograph Light enters the spectrograph in the form of parallel rays To achieve this illumination, a converging 3
This is particularly the case if the seeing is bad; then a narrow slit may let in only a fraction of the point source s light In fact to determine the spectral resolution of a spectrometer, let s consider a few things first We defined the dispersion of the grating as the variation of the reflection angle, δβ, with that of wavelength, δλ However, it is more useful to define it in terms of the change in the distance in the focal plane of the camera, δs Since in the small angle limit, β s/f cam, we just multiply the latter dispersion by the focal length of the camera, f c : s λ = 1 cosβ mf cam d Fig 4 The configuration of the spectrograph behind the focal plane of the telescope We see the collimator is placed such that its focal plane aligns with that of the telescope The grating or prism receives and scatters parallel light, which is focused onto the detector array by the camera Note, qualitatively, how a narrower slit can increase the spectral resolution lens, or collimator, is placed a distance of f c from the telescope s focal plane, where f c is the focal distance of the converging lens (Fig 4) 1 White light enters the spectrograph, and diverging bundles of light of the same wavelength leave the spectrograph at an angle of β A filter is used to select an order The bundles of parallel light are imaged or brought to a focus with a converging lens, called a camera, at a distance equal to that of the focal plane of the camera The camera forms multiple images of the source side-by-side at different wavelengths The spectral resolution of the spectrograph depends not only on the angular dispersion, which can be increased by working at a higher orders or by using a more narrow grooved spectrometer, but also on the entrance slit By narrowing the entrance slit, one decreases the overlap in the adjacent color images of the object, thereby increasing the spectral resolution Note that narrowing the slit also decreases the amount of light that enters the spectrograph, and thus compromises the signal to noise ratio of the spectrum 1 Note that the focal ratio of the collimator and telescope must be the same so that the collimator lens is right size to capture the image The reciprocal, called the reciprocal linear dispersion, is also often used Now the slit (or optical fiber) of width w, subtends an angle θ = w/f col on the sky The projected width of the slit at the focal plane of the camera is: w = w f cam f col To see this: consider a slit subtending an angle of w/fcolin the sky Considernow that thereis noprism and just white light In effect, consider the path of one color of light This light subtends an angle of w/fcol, in the spatial direction Assume, like the figure above shows, light going through an infinitely thin slit Then all of the light, once it passes through the collimator, forms a parallel beam If the slit has a width, then the light after the passing through the collimator will not be parallel but instead it will have an angular extent The projection of this angular extent, θ, on the focal plane of the camera equals θ /f cam = w fcam f col Essentially the small angle approximation assumes a linear relationship between the angle (in radians) and the projection length of the angle at a distance f However this does not include the effective magnification that results from the different projected sizes of the grating as seen by the camera and the collimator Differentiating the grading equation we find that: (cos α dα + cos β dβ) = m d dλ For a monochromatic source (dλ = 0): r = dβ dα = cos α cos β, 4
seeing, from the reciprocal linear dispersion: Fig 5 Imagine that the pencil is the slit, which is infinitely narrow and placed at the focal plane of the lens Here the rays, after going through the lens, which we take for the collimator, are parallel Fig 6 Here the pencil defines the upper and lower boundaries of the slit, which now has a width The rays, after going through the lens, veer off at an angle θ from the focal axis, thereby projecting width of w fcam f col on the camera s focal plane which equals d col /d cam, the ratio of the diameters of the collimator and camera So in fact: w = w f camd col f col d cam = w f cam d cam D f, sincethefocalratioofthetelescope(f/d)equalsthat of the collimator or λ = λ s w = Dw dcos(β) fd cam m λ = dcos(β) w d col D m f d cam d col Here the first term is the angular dispersion, A, of the spectrometer The second term on the right φ= w f is the angle that the slit width (w) projects on the sky The third term is the magnification caused by the different projected sizes of the grating as seen by the collimator and the camera, r Therefore: λ = rφd Ad col And the spectral resolving power is: R = λ λ = λad col rφd Thus, while the spectral resolving power depends on the slit width (or seeing), contained in the term φ, it also depends on the grating spacing and order (from A) and the camera and telescope configuration (from r, D and d col ) Note that for a given order, m, the spectral resolving power does not depend on the wavelength, λ, because the angular dispersion, A, depends on λ 1 R = d col rφd sin(α) + sin(β) cos(β) For this reason, we talk about the resolving power, R, of a telescope + grating setting, rather than its spectral resolution, λ 13 Resolving Power The spectral resolution of a telescope and instrument configuration is generally specified in terms of the spectral resolving power rather than the angular or linear dispersions The spectral resolving power, defined as R = λ λ, has the advantage of being independent of the wavelength, and rather a function of the specifications of the dispersion element (eg grating) and the telescope We can estimate this value, limited by a slit or the 14 Configurations of Grating Spectrometers Note that the dispersion (wavelength) direction of a spectrograph on the camera s focal plane is perpendicular to the projection of the slit length, while the spatial direction runs parallel to the direction of the slit A longslit spectrometer enables one to gather spatially resolved spectra of the field in the slit An echelle spectrograph is a grating spectrograph with a blaze that is optimized to observe the higher orders (eg m=100, and so forth) where the angular spacing between the orders decreases to the point that they overlap A second, perpendicularly mounted dispersive element (grating or prism) is then inserted as 5
15 Fourier Transform Spectrometer Fig 7 Optical fibers positioned at specific points on focal plane on the multi-object optical spectrograph, Hectospec, on the MMT Robots position the fiberswithin300secondstowithinanaccuracyof 25 µm Each fiber has a diameter of 250 µm, subtending 15 arcsec on the sky a cross disperser into the beam path This order separator disperses light at right angles to that of the main grating This technique produces a stacked series of spectra, each of which covers only part of the spectrum Suchagratingallowsonetorecordabroad spectral region at high resolution Yet, a relatively bright source is needed An echelle spectrum of the Sun, taken by NOAOnext door, is shown on the front of these notes The color is just added later to make the point (I guess) that the cross-disperser, by separating the orders, essentially stacks the spectrum from the blue wavelengths down to the red wavelengths Integral field unit spectrographs or fiber-fed spectrographs use optical fibers to sample many specific points on the focal plane of the detector, thereby getting spectra of hundreds of objects in the same image (Fig 5) Multiple slit spectrographs also obtain multiple spectra from a single image A separate mask of slits is prepared for each image, depending on the object list Here the field of view is smaller than that of the integral field spectrographs; yet this technique benefits from the lack of throughput losses associated with the optical fibers A Fourier Transform Spectrometer takes in the light from a source over a broad wavelength range Similar to the Michelson-Morley experiment the light is split up by a beamsplitter One half beam of light travels a certain proscribed distance defined by a fixed mirror; the other half of the beam travels a distance defined by a movable mirror The two beams then meet and superimpose, with one beam delayed as a result of the different distance traveled, which is adjusted by moving the adjustable mirror By making many measurements of the combined beams created with many discrete positions of the movable mirror, the spectrum of the source can be built up It is simply the Fourier Transform of the combined beam spectra Let I(ν) be the intensity, at frequency ν=1/λ, of the astronomical source, that which we want to determine If p is the difference in the paths of the two beams, the combined beam is therefore: I(p,ν) = I(ν)[1+cos(2πνp)] The sum of all the measurements taken with different mirror positions, d, is I(p) = 0 I(p,ν)dν = 0 I(ν)[1+cos(2πνp)]dν This is a Fourier cosine transform Taking the inverse, we get the object s spectrum I(ν) = 4 0 [I(p) 1 I(p = 0)]cos(2πνp)]dp 2 These spectrometers allow one to efficiently obtain high spectral resolution data of bright sources 6