3.4.2 The Diffraction Grating

• A diffraction grating is a plate on which there is a very large number of parallel, identical, close-spaced slits
• When monochromatic light is incident on a grating, a pattern of narrow bright fringes is produced on a screen

Diagram of diffraction grating used to obtain a fringe pattern

• The angles at which the maxima of intensity (constructive interference) are produced can be deduced by the diffraction grating equation:

• Exam questions sometime state the lines per m (or per mm, per nm etc.) on the grating which is represented by the symbol N
• d can be calculated from N using the equation

Angular Separation

• The angular separation of each maxima is calculated by rearranging the grating equation to make θ the subject
• The angle θ is taken from the centre meaning the higher orders are at greater angles

Angular separation

• The angular separation between two angles is found by subtracting the smaller angle from the larger one
• The angular separation between the first and second maxima n₁ and n₂ is θ₂ – θ₁

Orders of Maxima

• The maximum angle to see orders of maxima is when the beam is at right angles to the diffraction grating
  • This means θ = 90^o and sin θ = 1

• The highest order of maxima visible is therefore calculated by the equation:

• Note that since n must be an integer, if the value is a decimal it must be rounded down
  • E.g If n is calculated as 2.7 then n = 2 is the highest order visible

• When light passes through the slits of the diffraction grating, the path difference at the zeroth order maximum is zero
• At the first order maxima, there is constructive interference, hence the path difference is λ
• Therefore, at the nth order maxima, the path difference is equal to nλ

Using this diagram and trigonometry, the diffraction grating equation can be derived

• Using trigonometry, an expression for the first order maxima can be written:

• Where:
  • θ = the angle between the normal and the maxima
  • λ = the wavelength of the light (m)
  • d = the slit separation (m)

• This means, for n = 1:

• Similarly, for n = 2, where the path difference is 2λ:

• Therefore, in general, where the path difference is nλ:

• A small rearrangement leads to the equation for the diffraction grating:

d sin θ_n = nλ

• Diffraction gratings are useful for separating light of different wavelengths with high resolution
• They are used in spectrometers to:
  • Analyse light from stars
  • Analyse the composition of a star
  • Chemical analysis
  • Measure red shift / rotation of stars
  • Measure the wavelength / frequency of light from a star
  • Observe the spectra of materials
  • Analyse the absorption / emission spectra in stars

• Diffraction gratings also play a role in x-ray crystallography
  • X-rays are directed at a thin crystal sheet which acts as a diffraction grating to form a diffraction pattern
  • This is because the wavelength of x-rays is similar in size to the gaps between the atoms
  • This diffraction pattern can be used to measure the atomic spacing in certain materials

Diffraction gratings are most commonly used in spectrometers. These devices play a crucial role in areas of physics such as atomic physics and astrophysics