# Observing Techniques: spectroscopy

## why astronomical spectroscopy? from home.achilles.net/~ypvsj/data/elements/ from http://www.astrosurf.com/buil/us/peculiar2/pcygni.htm from http://instruct1.cit.cornell.edu/courses/astro101/java/binary/binary.htm from http://leo.astronomy.cz/mizar/pickering.htm

## illustrative example: the prism spectrograph source: Steve Dearden (www.astrosurf.com/dearden)

LO is the collecting lens, illuminated by the source of light L. L1 is the collimator and L2 the focusing lens or imaging lens. S1 is the entrance slit and S2 is its projection on the detector. We'll use these names also as sizes: S1 is also the size of the entrance slit, LO is the size of the collecting lens, etc. In astronomy, LO is the telescope objective or primary mirror, and so would be perhaps 100 times larger than L1 or L2. And in astronomy, the focusing lens and associated detector etc is often called "the camera."

The prism is of course the element that separates the light into various wavelengths. The physical effect is called dispersion, the fact that the refractive index of the glass is a function of wavelength. At any given input angle the output angles for light of two wavelengths will differ. If the difference in output angle is for a difference in wavelength then we call the angular dispersion. The linear dispersion, which fixes the actual scale of your final spectrum, is given by the angular dispersion multiplied by the focal length of the optics used to image the spectrum onto the detector.

The prism spectrograph illustrates clearly why the incoming beam has to be "collimated" - that is, light from each individual source has to presented to the prism as (nearly) parallel rays. The angles at which the rays emerge from the prism carry the wavelength information. If the input angles were also to carry positional information then the two would become confused. For the same reason, the slit has to be kept narrow: if the slit is too wide then there will be non-parallel contributions to the light hitting the prism.

An important formula:

s2 = s1 (L2/L1)

i.e. the slit is magnified in the ratio of the focal lengths of the focusing lens and collimator lens. More precisely, the image of the slit at any given wavelength is magnified in this ratio. The linear dispersion, which is the separation between different colours as measured at the detector (expressed for example in mm/Angstrom) can also be made greater when L2 is increased for a fixed value of L1. But since the projected slit size and the linear dispersion both increase at the same rate, increasing L2 does not necessarily improve the result. In fact, making L2 smaller allows you to fit a greater range of wavelengths on the detector. Why would you not want to make L2 too small?

If different sources are separated in the direction perpendicular to the dispersion, no confusion occurs. This is why a one-dimensional "slit" can be used instead of a small aperture. The prism spectrograph as shown is one possible example of a slit spectrograph. In an astronomical context the slit isolates the radiation being analysed to a narrow rectangular area on the sky. In practice this might mean that a single star is isolated, or a narrow line across a galaxy.

One can do better by having a large mask which is perforated by holes or slits where interesting objects are located. Of course that area of sky must have been investigated before to locate the spectroscopic targets, and to determine exact astrometry so that the masks can be made. source: "Subaru's Infrared Eye" (press release 2006)

In some cases the slit is not needed. This is the case if the source planeobjective prism spectrograph. Here is an image of a number of spectra spread out over the detector, each spectrum originating from a single star: source: Steve Dearden (Bob Fosbury, ESO-ECF)

## important tool: the diffraction grating The prism in the previous sketch can be replaced by a diffraction grating. A diffraction grating consists of a substrate (usually a rectangular glass plate) on which a regular pattern of grooves ("facets") has been placed. The grooves can be scratches made by a diamond (as when a LP record is created) but the creation of such a "master grating" is difficult and expensive. Commercial gratings are replicated from the master. The spacing between the grooves, which has to be kept very constant for good results, is usually called "d", the grating constant. Typically it is expressed in micrometres or, for convenience, in grooves per mm (g/mm) or sometimes l/mm (lines per mm). The figure (and the caption) explain the geometry. Unlike a prism, the grating can be of the reflecting type or transmitting type. In the case of the reflection grating the grooves are usually aluminised to increase the efficiency with which the light is diffracted. In the case of the transmission grating the grooves would be more likely to have an anti-reflective coating. The principles are similar and we'll discuss only the reflection grating in detail. As we will see, for light hitting the grating at angle (measured from the grating normal - no need here to worry about the angles of the triangular facets!) constructive interference will take place only at certain angles (labelled 1, -1 etc). This is shown in the next frame. This series of images is taken from the Diffraction Grating Handbook (Palmer and Loewen, Milton Roy Company, 2nd edition, 1994) The Grating Equation can be derived from the diagram on the left. "d" is the the groove period. We want to see whether light reflected from one facet can be in phase with light reflected from an adjacent facet. The proper approach would be to be to use diffraction theory (see Kitchin secn 4.1) but a geometric approach gives the correct angles. We compare a ray which strikes an (arbitrary) point on one facet with a ray which strikes the next facet, at a point a distance d away from the first ray. In the sketch at left, the ray which is closest to the arrow has to travel an extra distance ) compared with the ray which is parallel to it. For constructive interference to occur, the path length difference for the outgoing rays has to compensate the phase shift (modulo 2 ). As can be seen, this will happen when where a sign convention has been adopted: positive angles as those on the same side of the normal as the incident light. So in the case shown in the diagram is negative. We can now finally write the Grating Equation in its usual form: The "diffraction order" m can adopt any integer value, but for a given value of there will be (real) solutions for only for certain values of m -- as indicated already in Fig II-1 (a). Note that, measured in the diffraction plane as shown in this diagram, the beam size will in general be different after diffraction. This is true except when m = 0.

The angular dispersion in a given geometry can be found by differentiating the grating equation with held constant. One obtains: Don't forget though that after and m have been chosen, is fixed by the grating equation!

## blazed gratings etc

Look again at the grating equation. For a given choice of diffraction order, input angle, and wavelength this equation tells you in which direction to expect the outgoing ray to appear. This is forced by the simple requirements of constructive interference. But look at fig II-I. The facets (as they are called) of the grating have a triangular profile defined by the angle between the plane of the grating and one of the facets. The top of the triangle is usually 90o so only one angle needs to be given. The angle is called the blaze angle and although it does not change it does change the efficiency with which the grating diffracts into the direction .
Calculation of the efficiency as a function of the various parameters is a complex, and the results are very different for the two polarizations of the light. Often, though, a maximum of efficiency occurs at a combination of angles such that: which I will call the scalar blaze condition. It is the condition that corresponds to specular (mirror-like) reflection from the facets and so intuitively it should represent a maximum.

## important application: measuring redshift

Using spectroscopy one can measure the recession velocity of a star, galaxy, or other object, if (for example) we can detect an emission line whose "rest wavelength" is known.
The redshift z of a spectral line for an object whose velocity is much less than c is given by: which is an approximation to the following formula: which must be used for objects with a speed relative to the observer which is comparable to c .

Concerning this webpage please write to:
email:ndouglas@astro.rug.nl