Spectroscopy & Spectrographs

Spectroscopy & Spectrographs Roy van Boekel & Kees Dullemond

Overview • Spectrum, spectral resolution • Dispersion (prism, grating) • Spectrographs • longslit • echelle • fourier transform • Multiple Object Spectroscopy

Spectroscopy: what do we measure? • Spectrum = the intensity (or flux) of radiation as a function of wavelength • “Continuous” sampling in wavelength (as opposed to imaging, where we integrate over some finite wavelength range) • Note: In practice, when using CCDs for spectroscopy, one also integrates over finite wavelength ranges – they are just very narrow compared to the wavelength itself: Pixel width Δν << ν • Sampling is continuous but the spectral resolution is limited by the design of the spectrograph • Spectrum in classical sense holds no direct spatial information. Many spectrographs allow retrieving spatial info in 1 dimension, some even in 2 (“integral field units”)

Spectral resolution • Smallest separation in wavelength that can still be distinguished by instrument, usually given as fraction of  and denoted by R: or alternatively useful, though somewhat arbitrary working definition

Basic spectrograph layout • a means to isolate light from the source in the focal plane, usually a slit • “collimator” to make parallel beams on the dispersive element • dispersive element, e.g. a prism or grating. Reflection gratings much more frequently used than transmission gratings • “Camera”: imaging lens to focus beams in the (detector) focal plane + detector to record the signal

Dispersion Splitting up light in its spectral components achieved by one of two ways: • differential refraction • prism • interference • reflection/transmission grating • fourier transform • (Farby-Perot)

Prism • Refractive index n of material depends on wavelength • Several approximate formulae exist to describe n(). “Sellmeier” equation is accurate over a large wavelength range and used by manufacturers of optical glasses: • Bi and Ci are empirically determined coefficients. With 3 terms the Sellmeier approximation is accurate to 1 part in ~510-6 in the whole optical and near-infrared range

Prism • general light path through prism: one can show that: • dispersion is maximum for a symmetrical light path • dispersion is maximum for grazing incidence. Corresponding top angle  depends on refractive index of material. E.g. ~74° for heavy flint glass • However: most light is reflected instead of refracted for grazing incidence. In practice, smaller  are used (60° and 30° are common choices)

Prism dispersion curve strongly non-linear, dispersion in blue much stronger than in red part of spectrum

Prism spectrograph layout Credit: C.R. Kitchin “Astrophysical techniques” CRC Press, ISBN 13: 978-1-4200-8243-2

Young’s double slit experiment double slit lens screen incident wave θ d θ

Young’s double slit experiment double slit lens screen Optical path difference: incident wave Phase difference: d ΔP θ Add the two waves: Intensity is amplitude-squared:

Young’s double slit experiment N=2

Now a triple slit experiment... triple slit lens screen Optical path difference: incident wave Phase difference: d ΔP θ Add the three waves, and take the norm:

Now a triple slit experiment... N=3

Adding more slits... N=4

Adding more slits... 0th order 1st order 2nd order N=16

General formula of pattern N=4 Exercise: Show that the 1/N2 normalization is correct.

Width of the peaks N=4 For one has

Width of the peaks N=4 For with one has

Width of the peaks N=4 Peak width is therefore: For with one has (Later: Relevance for spectral resolution)

Now do 3 different wavelengths 0th order 1st order 2nd order N=4 Green is here the reference wavelength λ. Blue/red is chosen such that its 1st order peak lies in green’s first null on the left/right of the 1st order.

Now do 3 different wavelengths 0th order 1st order 2nd order N=8 Keeping 3 wavelengths fixed, but increasing N

Now do 3 different wavelengths 0th order 1st order 2nd order N=16 Keeping 3 wavelengths fixed, but increasing N

Now do 3 different wavelengths 0th order 1st order 2nd order N=4 Green is here the reference wavelength λ. Blue/red is chosen such that its 1st order peak lies in green’s first null on the left/right of the 1st order.

Now do 3 different wavelengths 0th order 1st order 2nd order N=8 Green is here the reference wavelength λ. Blue/red is chosen such that its 1st order peak lies in green’s first null on the left/right of the 1st order. Spectral resolution:

Let’s look at the 2nd order 0th order 1st order 2nd order N=8 Green is here the reference wavelength λ. Blue/red is chosen such that its 1st order peak lies in green’s first null on the left/right of the 1st order. Spectral resolution:

Let’s look at the 2nd order m=4 m=0 m=1 m=2 m=3 N=8 Green is here the reference wavelength λ. Blue/red is chosen such that its 1st order peak lies in green’s first null on the left/right of the 1st order.

Let’s look at the 2nd order m=2 N=8 Zoom-in around 2nd order Green is here the reference wavelength λ. Blue/red is chosen such that its 1st order peak lies in green’s first null on the left/right of the 1st order.

Let’s look at the 2nd order m=4 m=0 m=1 m=2 m=3 N=8 Green is here the reference wavelength λ. Blue/red is chosen such that its 1st order peak lies in green’s first null on the left/right of the 1st order.

Let’s look at the 2nd order m=4 m=0 m=1 m=2 m=3 N=8 Green is here the reference wavelength λ. Blue/red is chosen such that its 2nd order peak lies in green’s first null on the left/right of the 2nd order. Spectral resolution:

Let’s look at the 3rd order m=4 m=0 m=1 m=2 m=3 N=8 Green is here the reference wavelength λ. Blue/red is chosen such that its 3rd order peak lies in green’s first null on the left/right of the 3rd order. Spectral resolution:

General formula m=4 m=0 m=1 m=2 m=3 N=8 Green is here the reference wavelength λ. Blue/red is chosen such that its mth order peak lies in green’s first null on the left/right of the mth order. Spectral resolution:

Building a spectrograph from this Place a CCD chip here Make sure to have small enough pixel size to resolve the individual peaks.

Overlapping orders m=4 m=0 m=1 m=2 m=3 N=8 Going to higher orders means higher spectral resolution. But it also means: a smaller spectral range, because the “red” wavelengths of order m start overlapping with the “blue” wavelengths of order m+1

Effect of slit width triple slit lens screen incident wave d w

Effect of slit width single slit lens screen incident wave As we know from the chapter on diffraction: This gives the sinc function squared: w

Effect of slit width triple slit lens screen At slit-plane: Convolution of N-slit and finite slit width. At image plane: Fourier transform of convolution = multiplication incident wave d w

Effect of slit width N=16 d/w=8

Grating • many parallel “slits” called “grooves” • Transmission gratings and reflection gratings • width of principal maximum (distance between peak and first zeros on either side): • “Blazing”: tilt groove surfaces to concentrate light towards certain direction  controls in which order m light of given  gets concentrated  Credit: C.R. Kitchin “Astrophysical techniques” CRC Press, ISBN 13: 978-1-4200-8243-2 blazed reflection grating

 blazed transmission grating Grating, spectral resolution • resolution in wavelength:

w -i  d Reflection grating with groove width w and groove spacing d

Basic grating spectrograph layout Credit: C.R. Kitchin “Astrophysical techniques” CRC Press, ISBN 13: 978-1-4200-8243-2

Basic grating spectrograph layout Note: The word “slit” is here meant with a different meaning: Not a dispersive element, but a method to isolate a source on the image plane for spectroscopy. From here onward, “slit” will have this meaning. Dispersive slit = groove on a grating. Credit: C.R. Kitchin “Astrophysical techniques” CRC Press, ISBN 13: 978-1-4200-8243-2

Longslit spectrum • Very basic setup: entrance slit in focal plane, with dispersive element oriented parallel to slit (e.g. grooves of grating aligned with slit) • 1 spatial dimension (along slit) and 1 spectral dimension (perpendicular to slit) on the detector • Spectral resolution set by dispersive element, e.g. Nm for grating. • Spectrum can be regarded as infinite number of monochromatic images of entrance slit • projected width of entrance slit on detector must be smaller than projected size of resolution element on detector, e.g. for grating: where s is the physical slit width and 1 is the collimator focal length • slit width often expressed in arcseconds: where F is the effective focal length of the telescope beam entering the slit spatial direction   

Example longslit spectrum • high spectral resolution longslit spectrum of galaxy • Continuum emission from stars, several emission lines from star forming regions in galaxy spatial direction  wavelength 

Gratings: characteristics • Light dispersed. If d~ w most light goes into 1 or 2 orders at given . Light of (sufficiently) different  gets mostly sent to different orders • Light from different orders may overlap (bad, need to deal with that!) • Spectral resolution scales with fringe order m and is nearly constant within a fringe order  ~linear dispersion (in contrast to prism!) • Gratings are often tilted with respect to beam. Slightly different expression for positions of interference maxima: or equivalently i is the angle between the grating and the incoming beam. This expression is called the “grating equation”

Spectroscopy & Spectrographs