    ## Diffraction gratings and spectroscopy.

Diffraction gratings allow optical spectroscopy. A grating is a set of equally spaced, narrow, parallel sources. A grating disperses light of different wavelengths to give, for any wavelength, a narrow fringe. This allows precise spectroscopy. This page supports the multimedia tutorial Diffraction.

The sketch at right shows (top) a light source illuminating a grating, with the dispersed image projected on a distant screen. The images below show the dispersion patterns made with the same grating for a sodium lamp, a mercury lamp, an incandescent lamp and a candle. More details below.

### Two, three, four and many slits

To understand the pattern produced by a diffraction grating with many slits, let's begin with Young's experiment (two slits) and add more. In each of the animations below, we see at left the phasor diagram for the appropriate number of sources. The distance between the two ends of the figure is the amplitude of the resultant pattern, and the intensity, which is graphed at right, is proportional to the square of the amplitude. As the angle on the diffraction pattern is varied, the angle between the phasors varies and, at the same time, the black vertical line scans, in synchrony, across the plot of intensity. In each case, the angle between adjacent phasors scans from 0 to 2π.
 At left: phasor addition for two, three and four slits. At right: plot of intensity as the angle is varied. The vertical cursor on the intensity plots is in step with the phasor diagrams.

In Diffraction, we saw that the phase difference at angle φ θ between rays from two sources distance a apart was φ = 2πa sin θ/λ. With two slits, as φ varies from 0 to 2π, the phasor sum rotates so that its amplitude goes from maximum to zero to maximum. With three slits having the same spacing, the same varation in θ and thus φ takes us from central maximum, to zero at φ = 2π/3, to a small subsidiary maximum at φ = 2π/2 = π, to zero at φ = 4π/3 and back to a maximum at φ = 2π. For four slits, the first zero occurs at φ = π/2 and there are two small subsidiary maxima.

Consider the width of the large maximum, i.e. the spacing between the zeros on either side. For two slits, the range of φ is 2π = 4π/2; for three slits it is 4π/3, for four slits it is 4π/4 and, for N slits it is 4π/N. Note also that the height of the maxima increases, because more slits contribute to it. Remember that the intensity goes as the square of the amplitude, so the intensity of the bright fringes represented in the diagrams above would go as 22 : 32 : 42 = 4 : 9:  16 = 1 : 2.25 :  4.

So, as we increase the number of slits, the width of the fringes becomes narrower, their brightness increases, and the subsidiary maxima are proportionally less important. So let's now look at fringes with large numbers of slits. In all cases, if the slit separation is d, the condition for a strong maximum is the same as for Young's experiment, i.e.

d sin θm  =    where m is the order of the fringe.
Instead of specifying the interslit spacing d, we normally cite the number of slits per unit length, n. This gives fringes at
sin θm  = nmλ  where m is the order of the fringe.

### Diffraction gratings Diffraction gratings with 100, 200 and 300 lines/mm

Viewed in white light, we see the dispersion of different wavelengths for these three gratings.

### Patterns with monochromatic and broad band sources

 Gratings with different line spacings illuminated with monochromatic light from a laser and then broad band light from an incandescent lamp.

The three gratings shown above are first illuminated successively with monochromatic (red) laser light. Note the successively greater dispersion: the first order fringe of the grating with 300 lines/mm occurs at the same angle as the third order fringe of the grating with 100 lines/mm. Then the same three gratings are illuminated with an incandescent lamp, which emits a continuous spectrum of wavelengths.

Note the central bright fringe: for all wavelengths, this occurs at θ = 0, so this fringe is white. For the other fringes, sin θm = nmλ, where m is the order of the fringe and n is the number of fringers per unit length.

### Emission spectra

 Emission spectra

Now let's illuminate the grating with more complicated light sources. The central line is the undeflected beam of undispersed light at angle zero. The first order diffraction pattern appears on either side. A pure gas, when heated, emits light with specific wavelength (and thus specific energy per photon). Sodium and Mercury vapour lamps are viewed here through a grating. The incandescent lamp emits light from a hot metal filament with a continuous spectrum.

### Absorption spectra  Absorption spectrum of hydryogen Absorption spectrum of helium

A pure gas absorbs light with specific wavelength (and thus specific energy per photon). When light with a continuous spectrum of wavelengths passes through a gas, black lines on the transmitted spectrum show the wavelengths that have been absorbed: an absorption spectrum. The pictures show the absorption spectra for hydrogen and helium. These gasses are the principal components of the sun and its atmosphere, Helium (named for the Greek word for the sun) is the only element that was not discovered on Earth: the presence of helium in the solar atmosphere was deduced from the theoretical prediction of the absorption lines for the element with two protons in the nucleus and the observation of those absorption lines in the spectrum of light from the sun.

### Atomic energies and spectra Representations of the wave functions of the hydrogen atom

Why does a gas emit or absorb only discrete wavelengths, corresponding to photons with discrete energies? According to quantum mechanics, an electron in an atom can have only discrete values of energies, each energy corresponding to one particular orbital, described by a wave function. The diagram depicts the first several of these for hydrogen (the chance of interacting with an electron roughly corresponds to values of the square of the wavefunction, which is coded here as the brightness).  This work is licensed under a Creative Commons License.