# 2.1.3 Reflective diffraction gratings

Although the above description of diffraction has been in terms of light passing through a series of slits in a (transmission) diffraction grating, the type of grating which is currently most common in astronomy is a **reflective diffraction grating** or **reflection grating**. This again exploits the wave properties of light, in this case by making adjacent sections of a wavefront travel extra distances as it is reflected off a non-uniform surface. The non-uniform surface is actually a very precisely made mirror into which steps or grooves have been cut; as shown by the cross-section in Figure 20. The wavefront propagating from groove A and the wavefront propagating from groove B will constructively interfere with each other only if the difference in the lengths of the light paths, from L to L″, is an integer number of wavelengths. From the figure, the path difference is *d* sin *α* + *d* sin *β*, and the condition for constructive interference can therefore be written

This is such an important equation for astronomers that it is given a name, the **grating equation**. The integer *n* is called the **spectral order**, and quantifies how many wavelengths of path difference are introduced between successive grooves on the grating.

Now consider the grating equation. The groove spacing *d* is a feature of the grating, and the angle of the incident light *α* will be the same for all wavelengths, so the only remaining variables are the diffraction angle *β* and the wavelength *λ*. It is therefore clear that *β* must depend on wavelength, which is to say that the grating is a means of sending light of different wavelengths in different directions, i.e. producing a spectrum.

## ITQ 11

Imagine you have a grating spectrograph whose grating has 1000 grooves per mm, and is set up with the light incident at an angle of 15° to the grating normal. Calculate the angles at which light of: (i) 400 nm, (ii) 500 nm, and (iii) 600 nm will be diffracted in the first spectral order. You may find it convenient to express the wavelength and the groove spacing *d* in units of microns.

### Answer

Rearrange the grating equation *d*(sin *α* + sin *β*) = n*λ* and write

Then substitute in the values *d* = 0.001 mm = 1 μm, *α* = 15° and *n* = 1 to give sin *β* = *λ*/μm − 0.2588. We can then calculate the diffraction angles *β* as follows:

One feature of the spectrum produced by a diffraction grating is that multiple spectra are produced, corresponding to different spectral orders. For example, it is obvious from the grating equation that for a given spectrograph set-up, i.e. for some particular values of *d* and *α*, light at 700 nm in the first spectral order (*n* = 1) travels at the same diffraction angle *β* as light at 350 nm in the second spectral order (*n* = 2). Depending on the sensitivity of the detector and the relative flux in the source at overlapping wavelengths, it may be necessary to use a filter to block out the unwanted wavelengths.

It is instructive to ask how the choice of grating and spectral order affects the dispersion of the spectrum, i.e. the amount by which the light is spread out. The **angular dispersion** is a measure of how large a change Δ*β* in the diffraction angle results from a change δ*λ* in wavelength, so we want to know Δ*β*/Δ*λ*. Calculus makes the calculation of Δ*β*/Δ*λ* very straightforward, so we shall use that approach here. (If your calculus is too rusty for you to follow this, then skip the steps and just note the result.) The grating equation can be rearranged as sin *β* = *nλ*/*d* − sin *α*. Using calculus, we can then write

This indicates that the angular dispersion can be increased by working in higher spectral orders, i.e. by increasing *n*, or by using gratings with narrower groove spacings *d* (i.e. more grooves per millimetre such that *d* is smaller). Hence a grating with 600 grooves per millimetre will have twice the dispersion of a grating with 300 grooves per millimetre, if they are both used in the same spectral order. Of course, a grating of 300 grooves per millimetre used in second order (*n* = 2) will give the same dispersion as a grating with 600 grooves per millimetre used in first order (*n* = 1).

In the last few years, a new type of diffraction grating has become common in astronomy. The reflective diffraction grating described above works by introducing a different path length between parts of a wavefront striking different grooves of the grating. A **volume phase holographic grating** (VPH grating), in contrast, is a transparent medium, usually a layer of gelatine sandwiched between two glass plates. The refractive index of the gelatine varies in a carefully defined way from point to point. VPH gratings can offer superior efficiency and versatility to reflection gratings, and can be produced in the much larger sizes needed for the next generation of large telescopes.